computability theory and algebra · 2020. 10. 28. · jump on wtt-degrees, and proved an analogue...

Computability Theory and Algebra

Wu Huishan

School of Physical and Mathematical Sciences

A thesis submitted to the Nanyang Technological Universityin partial fulfilment of the requirement for the degree of

Doctor of Philosophy

2017

Acknowledgments

I would like to thank my supervisor, Prof. Wu Guohua, for teaching me com-

putability theory and guiding my research in the past few years. His knowledge,

suggestions, and discussions inspire me a lot, and I appreciate him for sharing me

with his ideas. His style of teaching, writing of papers and presentations influence

me a lot. I am grateful to him for all his efforts for guiding me to be a teacher and

to be a researcher. I also want to thank him for supporting my PhD study by his

research fund from MOE of Singapore.

I would like to thank Prof. Ng Keng Meng, for teaching me a course on first-

order logic. I also want to thank my peers and friends, Wang Shaoyi, Ru Junren,

Yu Hongyuan, Yuan Bowen for many inspiring discussion and various seminars on

logic and computability theory.

Last, but not the least, I want to thank my parents for their love and support.

i

Contents

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

1 Introduction 1

1.1 Basics of computability theory . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Part I: Bounded-jump operator and the high/low hierarchy . . . . . . 8

1.3 Part II: Degrees of orders on torsion-free abelian groups . . . . . . . . 11

1.4 Part III: Reverse mathematics, abelian groups and modules . . . . . . 17

I Bounded-jump operator and the high/low hierarchy 26

2 A bounded-low c.e. set which is low, but not superlow 27

2.1 Requirements and strategies . . . . . . . . . . . . . . . . . . . . . . . 28

2.2 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.3 Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.4 A bounded-low set with Turing degree 0′ . . . . . . . . . . . . . . . . 35

3 A bounded-high set which is high, but not superhigh 37

3.1 Pseudo-jump inversion via bounded-high sets . . . . . . . . . . . . . . 38

3.1.1 A Re-strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.1.2 A Pe-strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.1.3 A Qe-strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.1.4 Define the computable function gr . . . . . . . . . . . . . . . . 48

3.1.5 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.1.6 Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.2 Pseudo-jump inversion via bounded-low c.e. sets . . . . . . . . . . . . 60

iii

II Degrees of orders on torsion-free abelian groups 63

4 Group-order-computable degrees and PA degrees 64

4.1 Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.3 The P-strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.4 The Q-strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.5 The R-strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.5.1 Background on PA-complete sets . . . . . . . . . . . . . . . . 73

4.5.2 The R′-strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.6 Construction and verification . . . . . . . . . . . . . . . . . . . . . . 82

4.6.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.6.2 G is a computable group possessing computable orders . . . . 82

4.6.3 A-computable orders on G which are not computable . . . . . 84

4.6.4 Orders on G which are computable from PA-complete sets . . 90

5 Group orders and c.e. degrees 91

5.1 Requirements and strategies . . . . . . . . . . . . . . . . . . . . . . . 91

5.1.1 A Pe-strategy with permitting . . . . . . . . . . . . . . . . . . 92

5.1.2 The Q0-strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.1.3 The Q1-strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.1.4 The Q-strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.1.5 A Ne-strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.2 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.3 Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

III Reverse mathematics, abelian groups and modules 113

6 Reverse mathematics and divisible abelian groups 114

6.1 Divisible subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.2 Two decomposition theorems . . . . . . . . . . . . . . . . . . . . . . 118

iv

7 Reverse mathematics and modules 124

7.1 Projective modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

7.1.1 Direct sums of projective modules . . . . . . . . . . . . . . . . 124

7.1.2 Free modules and projective modules over Σ01-PIDs . . . . . . 127

7.2 Injective modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

7.2.1 Baer’s criterion for injective modules . . . . . . . . . . . . . . 129

7.2.2 Divisible modules and injective modules over Σ01-PIDs . . . . . 132

References 135

v

Abstract

This thesis mainly focuses on classical computability theory and effective aspects

of algebra. In particular, we will work on bounded low/high sets, degrees of orders

on torsion-free abelian groups, and reverse mathematics of several classic results in

modules.

In Part I, we will study bounded-low sets and bounded-high sets in terms of the

high/low hierarchy. Anderson and Csima proposed in [1] the notion of bounded-

jump on wtt-degrees, and proved an analogue of Shoenfield’s jump inversion theo-

rem. In [2], Anderson, Csima and Lange compared the bounded jump with Turing

jump and proved the existence of high bounded-low sets and low bounded-high sets.

Observing that superlow sets are all bounded-low, Anderson, Csima and Lange asked

in [2] whether there exist bounded-low c.e. sets which are low but not superlow,

and whether there exist superhigh sets which are not bounded-high. In Part I, we

will provide positive answers to these questions. We will also prove that there are

bounded-high sets which are high but not superhigh, and that there are bounded-low

c.e. sets which are high but not superhigh.

In Part II, we study Turing degrees of orders on computable torsion-free abelian

groups with infinite rank. Recently, Kach, Lange and Solomon [3] built a noncom-

putable c.e. set C and a computable torsion-free abelian group G with computable

orders such that every C-computable order on the constructed group is computable.

Motivated by this result, we call a degree a group-order-computable if there is a com-

putable torsion-free abelian group G with infinite rank which admits computable

orders such that every a-computable order on G is computable; in this case, say a is

group-order-computable via G . Following this definition, we call a degree a weakly

group-order-computable if there is a computable torsion-free abelian group G with

vi

infinite rank such that every a-computable group order on G is computable. We

will prove that a Turing degree a is group-order-computable iff a is weakly group-

order-computable iff a is not a PA degree. In particular, all c.e. degrees except 0′

are group-order-computable.

The objective to study group-order-computable degrees is indeed to explore de-

grees of orders on computable torsion-free abelian groups, because for a nonzero

degree a, if it is group-order-computable via G , then G has no orders of degree

a. We show that for any nonzero c.e. degree a, there is a computable torsion-

free abelian group G and a nonzero c.e. degree c < a such that G has orders of

degree ≥ a and c is group-order-computable via G , which means that G has no

incomputable orders of degree ≤ c.

In Part III, we study the reverse mathematics of classic results of divisible abelian

groups and modules. We will prove that over RCA0, the decomposition theorem of

divisible abelian groups and the decomposition theorem of torsion abelian groups

are both equivalent to ACA0. We then consider projective modules and injective

modules over Σ01-principal ideal domains (Σ0

1-PIDs). We will also show that ACA0

proves the following theorems:

• every projective module over a Σ01-PID is free;

• every submodule of a projective module over a Σ01-PID is projective;

• every divisible module over a Σ01-PID is injective;

• every quotient of an injective module over a Σ01-PID is injective.

vii

Chapter 1

Introduction

1.1 Basics of computability theory

We first give a brief introduction of basic concepts and theorems in computability

theory. Say that a function f with domain X ⊆ N is partial computable if there

is a Turing program P such that for x ∈ X, P running on input x stops after

finitely many steps, with output f(x); and for x ∈ X, P running on input x never

stops. A subset of natural numbers is computably enumerable (c.e. for short) if it

is the domain of a partial computable function. A subset A of natural numbers is

computable if both A and its complement are computably enumerable.

Turing programs can be effectively listed, i.e. via their Godel numbers, as Pe :e ∈ N. Let φe be the partial computable function computed by the e-th Turing

program Pe, and We be the domain of φe. Then φe : e ∈ N and We : e ∈ Nare effective enumerations of partial computable functions and c.e. sets respectively.

Similarly, we have an effective listing φke : e ∈ N of all k-ary partial computable

functions with k ≥ 2.

Lemma 1.1 (Padding Lemma) For each partial computable function φe, we can

effectively find an index e′ > e such that φe′ = φe.

Theorem 1.1 (Parameter Theorem) For a partial computable binary function, say

g, there is an injective computable function s such that for all x, y, φs(x)(y) = g(x, y).

Theorem 1.2 (Kleene’s Recursion Theorem) For any total computable function

f : N → N, there is an e0 such that We0 = Wf(e0).

1

Chapter 1. Introduction

Theorem 1.3 (Recursion Theorem with parameters, Kleene) For any total com-

putable function f : N2 → N, there is a computable function g : N → N such that

for any n, Wg(n) =Wf(n,g(n)).

Similar to Turing programs, all Turing programs with oracles can be effectively

listed, and we have a standard enumeration Φe : e ∈ N of partial computable

Turing functionals. For any set X and number x, we write ΦXe (x) ↓= y if y is the

output of the program Φe with oracle X and input x. The use of this computation

ΦXe (x), denoted by ϕXe (x), is defined as the greatest number z such that the question

z ∈ X or not is queried.

Theorem 1.4 (Parameter Theorem for functionals) For any partial computable bi-

nary functional, say Θ, there is an injective computable function s such that for any

oracle Y , and for all x, y, ΦYs(x)(y) ≃ ΘY (x, y).

Theorem 1.5 (Recursion Theorem for functionals) For any total computable func-

tion f : N2 → N, there is a computable function g : N → N such that for any oracle

Y and number n, ΦYg(n) = ΦY

f(n,g(n)).

Say that a set A is Turing reducible to B, written as A ≤T B, if there is a

Turing program with oracle B that outputs membership of A, and that A is Turing

equivalent to B, written as A ≡T B, if A ≤T B and B ≤T A. ≡T is an equivalence

relation over subsets of N, and we call the equivalence class of A the Turing degree

of A, denoted by deg(A). That is, deg(A) = B ⊆ N : B ≡T A. We also use a to

denote the Turing degree of A. ≤T gives rise to a natural order ≤ among Turing

degrees, i.e., a ≤ b if there are sets A ∈ a and B ∈ b such that A ≤T B. 0 denotes

the Turing degree of computable sets. If a degree a contains c.e. sets, then we call

this degree a c.e. degree.

In computability theory, we also consider strong reducibilities, like one-one re-

ducibility (≤1), many-one reducibility (≤m), truth-table reducibility (≤tt) and weak-

truth-table reducibility (≤wtt). Under thse strong reducibilities, we can define cor-

responding degrees:

2


• many-one degree (m-degree, for short) of A is defined as X ⊆ N : X ≡m

A. Here, X is many-one reducible to Y , denoted by X ≤m Y , if there is a

computable function f such that for all x, x ∈ X iff f(x) ∈ Y . Moreover, if f

is one-to-one, X is said to be one-one reducible to Y , denoted by X ≤1 Y .

• truth-table degree (tt-degree, for short) of A is defined as X ⊆ N : X ≡tt A.Let σn : n ∈ N be an effective list of all truth tables, i.e., propositional

formals obtained from atomic formals n ∈ X with n ∈ N. X is truth-table

reducible to Y , denoted by X ≤tt Y , if there is a computable function f such

that for all x, x ∈ X ⇔ Y σf(x).

• weak-truth-table degree (wtt-degree, for short) of A is defined as X ⊆ N :

X ≡wtt A. Here, X is weak-truth-table reducible to Y , denoted by X ≤wtt Y ,

if there is an oracle Turing program, say Φe, such that X = ΦYe with use ϕYe

recursively bounded (i.e., there is a computable function f such that f(x) ≥ϕYe (x) for all x). Weak-truth-table reducibility is also called bounded-Turing

reducibility.

Turing’s halting problem is one of the most fundamental results in computability

theory and has played a great role in the further development of this area.

Theorem 1.6 (Halting problem) K = e ∈ N : φe(e) ↓ is computably enumerable,

but not computable.

It is known that all c.e. sets are one-one reducible to K and Shoenfield’s limit

lemma shows that all sets Turing reducible to K have effective approximations.

The definition of Turing jump operator is based on a relativization of the Halting

problem. For a given set A, define A′ as the set e ∈ N : ΦAe (e) ↓, and we call A′ the

Turing jump of A. The Jump Theorem says that A ≤T B iff A′ ≤1 B′, so A ≡T B

implies A′ ≡T B′, and hence the jump operator defined above is well-defined. If a

is the degree of A, the degree of A′ is called the jump of a, and is denoted by a′.

The degree of Halting problem is just the Turing jump of 0, and denoted by 0′.

Obviously, if a is a c.e. degree, then a ≤ 0′. Friedberg’s jump inversion theorem

says that the Turing jump, as a mapping on Turing degrees, is surjective over 0′.

3


Let A(0) = A and A(1) = A′, the first jump of A. We then define by induction

the n-th jump of A, A(n), as the Turing jump of A(n−1). Let ∅ denote the empty

set. Below is the well-known high/low hierarchy in computability theory.

Definition 1.1 A set A ≤T ∅′ is called lown if A(n) ≡T ∅(n), and highn if A(n) ≡T

∅(n+1).

Definition 1.2 A degree c ≤ 0′ is lown if c(n) = 0(n), and highn if c(n) = 0(n+1).

low1, high1 are also called low, high respectively. So A ≤T ∅′ is low if A′ ≤T ∅′,

and high if A′ ≥T ∅′′. Mohrherr [4] named superlow sets and superhigh sets via the

tt-reducibility as: A ≤T ∅′ is superlow if A′ ≤tt ∅′, and superhigh if A′ ≥tt ∅′′.

Another hierarchy which is popular in computability theory is the Ershov hier-

archy, or the difference hierarchy.

Definition 1.3 For n ∈ N, a set A is called n-c.e., if there is a computable function

f(x, s) such that for all x,

• f(x, 0) = 0,

• A(x) = lims→∞

f(x, s),

• |s ∈ N : f(x, s) = f(x, s+ 1)| ≤ n.

A degree is n-c.e. with n ≥ 1 if it contains a n-c.e. set, and properly n-c.e. with

n ≥ 2 if it is n-c.e. but not (n − 1)-c.e. 1-c.e. sets (or degrees) are just c.e. sets

(or degrees), and 2-c.e. sets (or degrees) are often called d.c.e. sets (or degrees).

Cooper constructed a properly d.c.e. degree in 1971 in his PhD thesis [5], and his

constructions can be modified to show the existence of properly n-c.e. degrees for

n ≥ 2.

The set of all degrees is denoted by D, while the set of all c.e. degrees is denoted

by E . The following Figure 1.1 (from Cooper [6]) provides some basic information

of these two hierarchies.

Definition 1.4 A set A is called ω-c.e., if there is a computable function f(x, s)

and a computable function g such that

4


Figure 1.1: High/low hierarchy and Ershov hierarchy

• for all x, A(x) = limsf(x, s),

• for all x, |s ∈ N : f(x, s) = f(x, s+ 1)| ≤ g(x).

The following relations between ω-c.e. sets and strong reducibilities are well-

known:

Proposition 1.1 A set A is ω-c.e. iff A ≤tt ∅′ iff A ≤wtt ∅′.

The study of the structure of c.e. degrees stems from a question of Post, now

known as the Post’s problem:

Question 1.1 (Post (1944)) Does there exist c.e. degrees different from 0 and 0′?

In order to solve his problem, Post himself developed many important concepts

like simple sets, hypersimple sets and hyperhypersimple sets, with the attempt of

making the complements of the constructed c.e. sets not c.e., by making them

sparser and sparser. These concepts are closely related to strong reducibilities, in

the sense that even though these sets can be Turing complete, simple sets can never

be m-complete, and hypersimple sets can never be tt-complete.

Post’s problem was eventually solved by Friedberg in 1957 and Muchnik in 1956,

independently, where they introduced the injury method for the first time.

5


Theorem 1.7 (Friedberg (1957), Muchnik (1956)) There exist c.e. sets A and B

such that A T B and B T A.

A solution for Post’s problem, along Post’s original idea of making the comple-

ments sparse, was found in the 1970s, after Ershov’s work on η-maximal sets.

Friedberg and Muchnik’s construction has a feature that the injuries from higher

priority requirements are recursively bounded. In 1963, Sacks proved that any

nonzero c.e. degree splits and this construction has an inherited feature that the

injuries from higher priority requirements are finite, but cannot be bounded by any

computable function.

Theorem 1.8 (Sacks Splitting Theorem (1963)) Let B be a noncomputable c.e. set.

For any c.e. set A, there are low c.e. sets C and D such that

(1) C ∩D = ∅, A = C ∪D;

(2) B T C and B T D.

Theorem 1.8 says that any nonzero c.e. degree splits above 0. Robinson gener-

alized Sacks splitting theorem to the splitting of a given c.e. degree above a low c.e.

degree, which indicated that low c.e. degrees resemble degree 0.

Theorem 1.9 (Robinson Splitting Theorem (1971) [7]) Let L be a low c.e. set. For

any c.e. set A >T L, there are c.e. sets C and D such that

(1) C ∩D = ∅, A = C ∪D;

(2) A T C ⊕ L and A T D ⊕ L.

In 1964, Sacks proved that the c.e. degrees are dense, and the method used in

the proof is an infinite injury argument. That is, the injuries from higher priority

requirements can be infinite.

Theorem 1.10 (Sacks Density Theorem (1964)) For any two c.e. degrees a < b,

there is a c.e. degree c such that c ∈ (a,b).

6


The same style of construction was used by Sacks to prove the existence of

incomplete high c.e. degrees. In 1966, Lachlan introduced another kind of infinite

injury arguments in his proof of the existence of minimal pairs. Here, a pair of

nonzero c.e. degrees a and b form a minimal pair, denoted as a ∩ b = 0, if 0 the

infimum of a and b, i.e., is the only c.e. degree below both a and b.

Theorem 1.11 (Lachlan (1966), Yates (1966)) There are c.e. degrees forming a

minimal pair.

Cooper proved in 1974 that any high c.e. degree bounds a minimal pair, and in

this sense, high c.e. degrees resemble 0′.

Theorem 1.12 (Cooper (1974) [8]) Any high c.e. degree bounds a minimal pair of

c.e. degrees.

Miller proved in 1981 in his thesis [9] that any high c.e. degree bounds a minimal

pair of high c.e. degrees. This work was published by Slaman and Shore in their

paper [10], where a new construction was provided.

Definition 1.5 A c.e. degree c is cappable if it is zero or a half of a minimal pair,

that is, there is a c.e. degree a > 0 such that c∩a = 0; otherwise, c is noncappable.

It is easy to prove that every nonzero c.e. degree bounds a nonzero cappable

degree, which needs a nonuniform proof. The dual of cappable degrees is called

cuppable degrees.

Definition 1.6 (1) A c.e. degree c is cuppable if there is a c.e. degree a < 0′

such that c∪a = 0′; otherwise, c is called noncuppable, i.e., for any c.e. degree

a < 0′, c ∪ a < 0′.

(2) A c.e. degree c is called low-cuppable if there is a low c.e. a with c ∪ a = 0′.

Theorem 1.13 (Harrington (1976)) Any high c.e. degree bounds a high noncup-

pable degree.

7


Ambos-Spies, Jockusch, Shore and Soare proved in [11] that cappable degrees

form an ideal and noncappable degrees form a strong filter, providing a dichotomy

of c.e. degrees.

Theorem 1.14 (Ambos-Spies, Jockusch, Shore and Soare [11]) A c.e. degree is

noncappable iff it is low-cuppable (i.e., cups to 0′ via a low c.e. degree).

Lachlan proved in 1975 that Sacks splitting theorem and Sacks density the-

orem cannot be combined, where he developed a new technique, now known as

0′′′-arguments, a quite complicated construction of degrees.

Theorem 1.15 (Lachlan Nonsplitting Theorem (1975)) There are two c.e. degrees

a < b such that for any two c.e. degrees c,d, if b = c ∪ d, then b ≤ a ∪ c or

b ≤ a ∪ d.

Harrington later proved in his paper Understanding Lachlan’s monster paper

that Lachlan’s nonsplitting theorem can be strengthened as follows:

Theorem 1.16 (Harrington Nonsplitting Theorem (1980)) There exists a c.e. de-

gree a < 0′ such that 0′ cannot split above a.

1.2 Part I: Bounded-jump operator and the high/low

hierarchy

The notion of bounded-jump operator † was proposed by Anderson and Csima in

paper [1], where they tried to find an appropriate jump operator on wtt-degrees1.

For a set A, the bounded-jump of A is defined as the set

A† = e ∈ N : ∃i ≤ e[φi(e) ↓ & ΦAφi(e)e (e) ↓],

where A z:= x ∈ A : x ≤ z. Note that A† ≤T A ⊕ ∅′. In [1], Anderson and

Csima pointed out this bounded-jump operator † for wtt-reducibility behaves like

the Turing jump ′ for Turing reducibility, i.e., (1) ∅† and ∅′ are 1-equivalent, (2)

for any set A, A <wtt A†, and (3) for any sets A,B, if A ≤wtt B, then A† ≤wtt B

†.

1In paper [1], Anderson and Csima used b for the bounded-jump operator †.

8


(2) implies that A† ≡T A when A ≥T ∅′. In the same paper, Anderson and Csima

proved an analogue of Shoenfield’s jump inversion theorem for this bounded-jump

operator †.

Say that a set A is bounded-low if A† ≤wtt ∅†, and bounded-high if A ≤wtt ∅†

and ∅†† ≤wtt A†. There are several immediate consequences: bounded-low sets

and bounded-high sets are all ω-c.e.; bounded-low sets are closed downwards under

wtt-reducibility; and bounded-high sets are closed upwards under wtt-reducibility

on ω-c.e. sets. Furthermore, bounded-low sets cannot be bounded-high because

A† ≤wtt ∅† implies A† <wtt A†† ≤wtt ∅††.

It is easy to see that A is bounded-low if and only if A† is ω-c.e. Thus, all superlow

sets are bounded-low, because for all superlow sets A, A† ≤1 A′ ≤tt ∅′ ≡1 ∅†. Also

note that ∅′ ≡1 ∅† is bounded-high, because ∅†† ≤wtt (∅′)†.

Proposition 1.2 (Bickford and Mills (1982) [12]) There are two superlow c.e. sets

A,B such that A⊕B ≡T ∅′.

The proposition above shows that there are two bounded-low sets whose join is

bounded-high, thus not bounded-low. One natural question is:

Question 1.2 (1) Whether bounded-low sets are closed downwards under Turing

reducibility?

(2) Similarly, whether bounded-high sets are closed upwards under Turing re-

ducibility on ω-c.e. sets?

For a superlow A, if B ≤T A, then B′ ≤1 A

′ ≤tt ∅′. Thus B is also superlow.

Superlow sets are closed downwards under Turing reducibility. Similarly, superhigh

sets are closed upwards under Turing reducibility.

In Chapter 2, we will give negative answer to Question 1.2 by constructing a

bounded low c.e. set which Turing computes ∅′. This tells us that the greatest c.e.

Turing degree 0′ contains both bounded-low c.e. sets and bounded-high c.e. sets,

and one easy consequence is:

Proposition 1.3 (1) There are bounded-low c.e. sets which are superhigh.

9


(2) There are bounded-high c.e. sets which are superhigh.

Recently, in [2], Anderson, Csima and Lange initiated the study of comparing

the bounded-jump and the Turing jump, where they constructed high bounded-low

sets, and low bounded-high sets, showing that there is no direct relation between

these two jumps. Motivated by the fact that superlowness implies bounded-lowness,

they asked the following related questions:

Question 1.3 (Anderson, Csima and Lange [2])

(1) Does there exist a bounded-low set which is low but not superlow?

(2) Does there exist a bounded-high set which is high but not superhigh?

(3) Does there exist a superhigh set which is not bounded-high?

Question 1.3.(3) has positive answer because of the existence of bounded-low c.e.

sets which are superhigh.

Figure 1.2: Turing degrees of bounded-low sets

We will also provide positive answers to the other two questions. In chapter 2,

we will construct a bounded-low c.e. set which is low but not superlow directly.

In chapter 3, we show the existence of bounded-high sets which are high but not

superhigh by proving a pseudo-jump inversion theorem for bounded-high sets; we

will then prove a pseudo-jump inversion theorem for bounded-low c.e. sets, and this

implies the existence of bounded-low c.e. sets which are high but not superhigh.

10


Figure 1.3: Turing degrees of bounded-high sets

Finally, we point out here that all bounded-high sets we constructed are ω-c.e.,

and that∅′ is the only bounded-high c.e. set we know. It seems not easy to construct

bounded-high c.e. sets, the questions related to constructions of bounded-high c.e.

sets are still open as far as we know. For example, the following two questions still

open.

Question 1.4 (Anderson and Csima [1]) Whether the analogue of Sacks’s jump

inversion is true or not? That is, for any set A with ∅† ≤wtt A ≤wtt ∅††, does there

exist a c.e. set C ≤wtt ∅† such that C† ≡wtt A?

Question 1.5 (Anderson, Csima and Lange [2]) Does there exist a low c.e. set

which is bounded-high?

1.3 Part II: Degrees of orders on torsion-free abelian

groups

An abelian group G = (G,+G , 0G ) is orderable if there is a linear order ≤G on G

such that for all x, y, z in G, x ≤G y implies x +G z ≤G y +G z; say ≤G is a group

order (or simply order) on G . It is known that an abelian group is orderable iff it

is torsion-free (Levi’s theorem). So we will restrict ourselves to torsion-free abelian

groups.

From an order ≤G on G , one can form the set P≤G:= g ∈ G : g ≥G 0G of all

positive elements under ≤G . P≤Gis a positive cone of ≤G . On the other hand, if

11


X ⊆ G satisfies the following conditions: (1) X is a semigroup, (2) X∩−X = 0G ,

(3) X ∪ −X = G, where −X = −g : g ∈ X, then we can define an order ≤X by

x ≤X y ⇔ y − x ∈ X

and X is the positive cone of the order ≤X . Indeed, this entails a one-to-one

correspondence between orders and positive cones on G .

An abelian group G = (G,+G , 0G ) is computable if its domain G and addition

+G are both computable. Note that under this definition, the inverse function on G

is computable. If G is a computable torsion-free abelian group, we will use X(G ) to

denote the set of positive cones of G , and we will study Turing degrees of members

of X(G ). Note that for such a group G , P≤Gand ≤G are Turing equivalent.

Definition 1.7 A class C of sets is called a Π01 class if it can be represented as the

set of all infinite paths of some computable binary tree.

For a computable torsion-free abelian group G , the positive cones of G are defined

via Π01 conditions, and one can show that X(G ) can be represented as the set of all

infinite paths of a computable binary tree, which means that X(G ) is a Π01 class.

We are mainly interested in the degrees of members of X(G ), i.e., the degrees

of orders on G . Classical results on degrees of members of Π01 classes are due to

Jockusch and Soare [13] in 1972, where the low basis theorem and the hyperimmune-

free basis theorem are provided. These two basis theorems imply that every com-

putable torsion-free abelian group has orders of low degree, and of hyperimmune-free

degree, respectively.

Definition 1.8 For a torsion-free abelian group (G,+, 0G), g0, · · · gn ∈ G are lin-

early independent if for any a0, · · · an ∈ Z,

a0g0 + · · ·+ angn = 0G → a0 = · · · = an = 0.

An infinite subset B of G is linearly independent if any finite subset of B is linearly

independent. A maximal linearly independent set is called a basis of G.

12


For 1 ≤ k ≤ ∞, let Qk denote the direct sum of k many copies of additive

group Q. In particular, Q∞ = ⊕i∈N

Q, its elements are just finite Q-linear sums over a

computable basis ei : i ∈ N, where ei = (· · · , 1, 0, · · · ) is the sequence whose i-th

column is 1 and all other columns are 0.

The rank of G, rank(G), is defined as the size of a basis which is the least number

k ≤ ∞ such that G embeds into Qk. Here are some easy examples:

• rank(Z) = rank(Q) = 1.

• rank(Z⊕ Z) = rank(Q⊕Q) = 2.

• Let G be the subgroup ofQ⊕Q generated by (1, 0), (0, 1), (12, 12). Elements of

G are of the form m(1, 0)+n(0, 1)+k(12, 12) with m,n, k ∈ Z. Then rank(G) =

2.

We use O(G ) to denote the set of all degrees of orders on a computable torsion-

free abelian group G . Solomon [14] proved the following results about O(G ):

(1) if rank(G ) = 1, then O(G ) contains exactly one degree 0;

(2) if 2 ≤ rank(G ) <∞, then O(G ) contains all degrees;

(3) if rank(G ) = ∞, then O(G ) contains all degrees ≥ 0′.

For a computable torsion-free abelian group G and a Π01 class C, we say X(G )

degree represents C if

O(G ) = deg(C) : C ∈ C,

where deg(C) is the Turing degree of C. Jockusch and Soare in [13] constructed an

infinite Π01 class, say S, in which any two distinct elements are Turing incomparable.

Then for any computable torsion-free abelian group G , O(G ) = deg(C) : C ∈ S,because either O(G ) = 0 or O(G ) ⊇ d : d ≥ 0′.

It turns out that the Π01 class of positive cones on computable torsion-free abelian

groups cannot degree represent all Π01 classes. However, an earlier result of Metakides

and Nerode [15] in 1979 showed that any Π01 class can be degree represented by the

Π01 class of positive cones on some computably formally real field in a much stronger

manner.

13


For a computable torsion-free abelian group G with finite rank, we know that

O(G ) is either 0 or D, the set of all degrees. However, for a computable torsion-

free abelian group G with infinite rank, although O(G ) ⊇ d : d ≥ 0′, we don’t

know what the set O(G ) exactly is. Here are two related results:

• (Solomon [14]) If B is a basis of G , then d : d ≥ deg(B) ⊆ O(G ).

• (Kach, Lange, Solomon and Turetsky [3]) O(G ) must contain infinitely many

low degrees as well as infinitely many hyperimmune-free degrees.

In [3], Kach, Lange and Solomon considered the question whether O(G ) is up-

wards closed for a computable torsion-free abelian group G with infinite rank, and

provided a negative answer to it.

Theorem 1.17 (Kach, Lange and Solomon [3]) There is a computable torsion-free

abelian group G with infinite rank, and a noncomputable c.e. set C such that G has

exactly two computable orders and every C-computable order on G is computable.

G in Theorem 1.17 has no orders in deg(C), but has orders in 0. By deg(C) > 0,

O(G ) is not upwards closed.

Definition 1.9 A set C is called group-order-computable via G if G is a computable

torsion-free abelian group with infinite rank which admits computable orders, and

every C-computable order on G is computable. C is group-order-computable if it is

group-order-computable via some G .

Definition 1.10 A Turing degree c is called group-order-computable if it contains

a set C which is group-order-computable.

0 is obviously group-order-computable, and group-order-computable degrees are

downwards closed. On the one hand, Kach, Lange and Solomon’s theorem above

implies the existence of nonzero c.e. group-order-computable degrees, and hence

can be low. In his thesis [16], C. J. Martin showed the existence of high c.e. degrees

which are group-order-computable. On the other hand, any a ≥ 0′ cannot be group-

order-computable because every computable torsion-free abelian group with infinite

rank has an order of degree a.

Our main objective in chapter 4 is to investigate the characterization of group-

order-computable degrees.

14


Definition 1.11 A set C is said to be weakly group-order-computable via a com-

putable torsion-free abelian group G with infinite rank if every C-computable order

on G is computable. C is weakly group-order-computable if C is weakly group-order-

computable via some G .

A Turing degree is weakly group-order-computable if it contains a weakly group-

order-computable set. Clearly, group-order-computable degrees are weakly group-

order-computable. We will show that these two families of degrees, and also non-

PA-degrees, coincide.

A set is called PA-complete if it computes a complete extension of Peano Arith-

metic. An equivalent definition of PA-complete sets is that:

• a set A is PA-complete iff it computes a 0, 1-valued diagonally nonrecursive

function,

where a function f is diagonally nonrecursive if ∀x[φx(x) ↓→ f(x) = φx(x)]. A

degree is PA if it contains a PA-complete set. The notion of PA-degrees plays a

vital role in the study of randomness. For example, Stephan [17] proved that a PA

degree a is Martin-Lof random iff a ≥ 0′.

The following facts for PA-degrees are well-known:

Proposition 1.4 (Scott and Solovay [18]) Let a be a Turing degree. The following

are equivalent:

(1) a computes a complete extension of Peano Arithmetic.

(2) a computes a consistent extension of Peano Arithmetic.

(3) Each Π01 class contains an a-computable element.

We first characterize weakly group-order-computable degrees in terms of PA-

degrees.

Proposition 1.5 Let a be a Turing degree. The following are equivalent.

(1) a is a PA degree.

15


(2) a is not weakly group-order-computable.

Proof: (1) ⇒ (2). Let a be a PA degree, and G be a computable torsion-free

abelian group with infinite rank (not necessarily possessing computable orders). We

only need to show that G has an a-computable order which is not computable.

Martin proved in [16] that X(G ) has a Π01 subclass, say C(G ), containing no

computable orders. As a computes a positive cone in C(G ), G has an a-computable

order which is not computable.

(2) ⇒ (1). We first mention a result of Hatzikiriakou and Simpson [19] in reverse

mathematics: the statement “every countable torsion-free abelian group is orderable”

is equivalent to WKL0 over RCA0. This result has a consequence that there is a

computable torsion-free abelian group, say G0, with a property that all orders of it

are of PA degree. So G0 has no computable orders, hence the rank of G0 is infinite.

By definition, a degree a is not weakly group-order-computable, if any com-

putable torsion-free abelian group with infinite rank has an a-computable order

which is not computable. In particular, G0 has an a-computable order, and hence a

computes an order on G0, which is of PA degree. This implies that a itself is a PA

degree.

By Proposition 1.5, all PA-degrees are not group-order-computable. We will

construct in chapter 4 a computable torsion-free abelian group G with infinite rank

such that for any set A, G admits exactly two A-computable orders iff A is not

PA-complete.

As computable sets are not PA-complete, G above admits exactly two com-

putable orders. Then for any set A which is not PA-complete, every A-computable

order on G is computable; for any set A which is PA-complete, A computes a

noncomputable order on G . This shows that for any degree a, if it is now not

group-order-computable, then it is a PA degree. Hence, a degree is PA iff it is not

weakly group-order-computable iff it is not group-order-computable.

At the end of [3], by applying the low basis theorem of Π01 classes infinitely many

times, Kach, Lange, Solomon and Turetsky showed that every computable torsion-

free abelian group of infinite rank has orders of infinitely many low degrees. We

would like to know whether G in Theorem 1.17 has orders of other degrees:

16


Question 1.6 Given a nonzero c.e. degree a, can we build a group G such that:

(1) it has exactly two computable orders, (2) G has an order of degree a, and (3)

there is a noncomputable c.e. C such that every C-computable group order on G is

computable?

We will give a positive answer to Question 1.6 in chapter 5. Our results show

that for any nonzero c.e. degree a, there is a nonzero c.e. degree c < a, and a

computable torsion-free abelian group G such that G has orders of degree ≥ a but

has no incomputable orders of degree ≤ c.

1.4 Part III: Reverse mathematics, abelian groups

and modules

In this part, we study reverse mathematics of some classic theorems in module

theory. We first briefly introduce common subsystems of second order arithmetic

studied in reverse mathematics. L2, the language of second order arithmetic, con-

tains two-sorted variables, namely, number variables and set variables. The full

system Z2 of second order arithmetic contains basic axioms of ordered semiring of

natural numbers, induction axioms of sets, i.e., for any set X,

(0 ∈ X ∧ (n ∈ X → n+ 1 ∈ X)) → ∀n(n ∈ X),

and comprehension axioms of formulas φ(n) in L2, i.e.,

∃X∀n(n ∈ X ↔ φ(n)),

where X does not occur freely in φ.

Many theorems in non-set-theoretic mathematics are classified by five big sub-

systems of Z2, i.e., RCA0, WKL0, ACA0, ATR0 and Π11-CA0. Simpson’s book [20]

provides a comprehensive introduction of these five subsystems. We introduce RCA0

and ACA0 in detail here, as we will mainly work with these two subsystems.

RCA0 contains basic axioms of ordered semiring of natural numbers, Σ01-induction

and ∆01-comprehension. Σ0

1-induction is the axiom of the following form with φ re-

stricted to Σ01 formulas:

(φ(0) ∧ (φ(n) → φ(n+ 1))) → ∀nφ(n).

17


∆01-comprehension is the axiom of the following form with φ restricted to ∆0

1 for-

mulas:

∃X∀n(n ∈ X ↔ φ(n)),

where X does not occur freely in φ.

ACA0 contains basic axioms of ordered semiring of natural numbers, induction

axioms of sets, and Σ0k-comprehension for all k. Σ0

k-comprehension axioms say

∃X∀n(n ∈ X ↔ φ(n)),

where φ is a Σ0k-formula and X does not occur freely in φ(n).

The following theorem is widely used in reverse mathematics when proving a

theorem in ordinary mathematics implies ACA0.

Theorem 1.18 The following are equivalent over RCA0.

(1) ACA0;

(2) Let f : N → N be a one-to-one function, then the range of f exists.

We now define basic concepts of module theory using the language L2 of second

order arithmetic.

Definition 1.12 (RCA0) Let R be a commutative ring with identity 1R, a module

M over R is an abelian group together with a scalar operation from R×M to M ,

satisfying for all m,m1,m2 ∈M and r, r1, r2 ∈ R,

M1. (r1 + r2) m = r1 m+ r2 m;

M2. (r1r2) m = r1 (r2 m);

M3. 1R m = m;

M4. r (m1 +m2) = r m1 + r m2.

For convenience, we write rm for r m.

Definition 1.13 (RCA0) Let M1 and M2 be two R-modules. A map φ :M1 →M2

is a R-module homomorphism if for any x, y ∈M1 and r ∈ R, φ(x+y) = φ(x)+φ(y),

φ(rx) = rφ(x).

18


Definition 1.14 (RCA0) Let φ : A → B and ψ : B → C be two R-module homo-

morphisms. The sequence Aφ→ B

ψ→ C is exact at B if ker(ψ) = im(φ). Moreover,

if φ is a monomorphism and ψ is a surjective homomorphism, then the sequence

0 → Aφ→ B

ψ→ C → 0

is also exact at A, C, and often called a short exact sequence.

We only consider countable modules over a commutative ring with identity and

mainly deal with infinite modules whose domain are subsets of N. Within RCA0,

the set N<N of all codes of finite sequences of natural numbers exists (see [20] for

more details). The symbol ⟨x0, x1, · · · , xn⟩ denotes the unique code of a sequence

of natural numbers of length n+ 1, we often identify a sequence with its code.

For a sequence σ, lh(σ) denotes the length of σ. We use λ to denote the empty

sequence, and lh(λ) = 0. For a nonempty sequence σ, lh(σ) ≥ 1, and we often write

σ = ⟨σ(0), σ(1), · · · , σ(lh(σ)−1)⟩. For two sequences σ, τ , the concatenation σaτ of

them is just the sequence ⟨σ(0), σ(1), · · · , σ(lh(σ)−1), τ(0), τ(1), · · · , τ(lh(τ)−1)⟩.

Definition 1.15 (RCA0) For 0 ≤ i ≤ n, let (Mi,+Mi, Mi, 0Mi

) be a R-module.

The direct sum⊕

0≤i≤nMi of modules Mi(0 ≤ i ≤ n) is a R-module containing finite

sequences of length n+1 of the form ⟨m0,m1, · · · ,mn⟩ with each mi ∈Mi, together

with componentwise addition

⟨m0, · · · ,mn⟩+ ⟨m′0, · · · ,m′

n⟩ = ⟨m0 +M0 m′0, · · · ,mn +Mn m

′n⟩

for abelian groups and componentwise scalar multiplication

r⟨m0, · · · ,mn⟩ = ⟨r M0 m0, · · · , r Mn mn⟩

for modules. The zero of⊕

0≤i≤nMi is ⟨0M0 , 0M1 , · · · , 0Mn⟩. If Mi =M for all 0 ≤ i ≤

n, we often use Mn+1 to denote⊕

0≤i≤nM .

In general, as in Solomon [21], we can define the direct sum of countably many R-

modules as a module whose nonzero elements are finite sequences with last column

nonzero, and zero element is just the empty sequence.

19


Definition 1.16 (RCA0) Let Mi(i ∈ N) be R-modules. The direct sum⊕i∈NMi is

a R-module containing finite sequences of the form ⟨m0,m1, · · · ,mn⟩ with n ∈ N,mi ∈ Mi for 0 ≤ i ≤ n, mn = 0Mn, and the empty sequence, together with compo-

nentwise addition and componentwise scalar multiplication. Empty sequence is the

zero element of⊕i∈NMi. We simply write

⊕i∈NM (or M∞) for

⊕i∈NMi when Mi = M

for all i ∈ N.

Definition 1.17 (RCA0) A short exact sequence 0 → Aµ→ B

ε→ C → 0 splits if

there exists a homomorphism σ : C → B such that εσ is the identity homomorphism

on C. σ is called a splitting for this short exact sequence.

Lemma 1.2 (RCA0) If the short exact sequence 0 → Aµ→ B

ε→ C → 0 splits with

a splitting σ : C → B, then B ∼= A⊕ C. Moreover, B = µ(A)⊕ σ(C).

Proof: Define a map ψ : A⊕ C → B; ⟨a, c⟩ 7→ µ(a) + σ(c). We now show that ψ

is an isomorphism between A⊕ C and B.

For any a, a′ ∈ A, c, c′ ∈ C, and r ∈ R,

ψ(⟨a, c⟩+ ⟨a′, c′⟩) = ψ(⟨a+ a′, c+ c′⟩)

= µ(a+ a′) + σ(c+ c′)

= µ(a) + µ(a′) + σ(c) + σ(c′)

= ψ(⟨a, c⟩) + ψ(⟨a′, c′⟩)

and

ψ(r⟨a, c⟩) = ψ(⟨ra, rc⟩) = µ(ra) + σ(rc) = rµ(a) + rσ(c) = rψ(⟨a, c⟩).

So ψ is a R-module homomorphism.

ψ is injective. First, ψ(⟨a, c⟩) = µ(a) + σ(c) = 0 implies 0 = ε(µ(a) + σ(c)) =

0 + εσ(c) = c. Then µ(a) = 0, and this implies a = 0. So ⟨a, c⟩ = 0.

ψ is surjective. For any b ∈ B, ε(b) ∈ C and ψ(⟨0, ε(b)⟩) = σε(b), εσε(b) = ε(b).

So σε(b)−b ∈ ker(ε) = im(µ). Then there exists an a ∈ A such that µ(a) = σε(b)−b,and then b = µ(a) + σε(b) = ψ(⟨a, ε(b)⟩). This also implies that B = µ(A) + σ(C).

Now we show B = µ(A)⊕ σ(C). It is enough to prove µ(A) ∩ σ(C) = 0. Letb ∈ B. Assume b = µ(a) = σ(c) for some a ∈ A and c ∈ C. Then ψ(⟨a,−c⟩) =

20


µ(a)− σ(c) = 0. Similarly, 0 = ε(µ(a)− σ(c)) = 0− εσ(c) = −c. Then c = 0, and

thus b = 0.

In Chapter 6, we first consider reverse mathematics of divisible abelian groups.

Definition 1.18 (RCA0) Let G be an abelian group, and g ∈ G.

(1) g is torsion if there is a natural number n ≥ 1 such that ng = 0G.

(2) The order o(g) of an element g ∈ G is the least natural number n ≥ 1 such

that ng = 0G if g is torsion, and ∞ otherwise.

An abelian group G is torsion if any element of G has a finite order, and torsion-

free if any nonzero element of G has an infinite order. Notions like torsion groups,

torsion-free groups are all definable within RCA0.

Definition 1.19 (RCA0) An abelian group G is divisible if for every g ∈ G and for

every nonzero n ∈ N, there is a g′ ∈ G such that g = ng′.

For a prime number p, Z(p∞) = npm

: m,n ∈ Z/Z exists in RCA0.

We will prove that over RCA0, the following are equivalent:

(1) ACA0.

(2) Every divisible subgroup of an abelian group is a direct summand.

(3) Every torsion-free abelian group has the largest divisible subgroup.

(4) Every divisible abelian group is isomorphic to a direct sum of the form⊕n∈I

(Z(p∞n ))ln ⊕⊕n∈J

Q for some I, J ⊆ N, and if I = ∅, for each n ∈ I,

1 ≤ ln ≤ ∞.

For an R-module M and a subset A of M , the set

⟨A⟩ = r0a0 + · · ·+ rnan : n ∈ N, 0 ≤ i ≤ n, ri ∈ R, ai ∈ A

of R-linear combinations of elements of A forms a submodule of M . If M = ⟨A⟩,we say A generates M or A is a set of generators of M .

A finite set m0,m1, · · · ,mn of M is linearly independent if for any ri ∈ R, 0 ≤i ≤ n, r0m0 + · · · + rnmn = 0 implies ri = 0 for all 0 ≤ i ≤ n. An infinite subset

A of M is linearly independent if any finite subset of A is linearly independent. A

R-module M has a basis B if B is linearly independent and B generates M .

21


Definition 1.20 (RCA0) A R-module is free if it has a basis.

Proposition 1.6 (RCA0) Let F be a free R-module with an infinite basis, say B =

bi : i ∈ N. Then F is isomorphic to⊕i∈NR.

Proof: Let ci = ⟨0, · · · , 0, 1R⟩ ∈⊕i∈NR of length i + 1. That is, the j-th column is

the zero element of R for j ≤ i− 1 and the i-th column is the identity of R. Then

ci : i ∈ N is a basis of⊕i∈NR. Define φ : F →

⊕i∈NR by φ(

∑0≤i≤n

ribi) =∑

0≤i≤nrici for

any n ∈ N and ri ∈ R with 0 ≤ i ≤ n.

Proposition 1.7 (RCA0) Let M be a R-module. Then M is isomorphic to a quo-

tient of the free module F =⊕i∈NR.

Proof: List M as m0,m1, · · · . Define a map φ : F →M as follows: first, for all i,

φ(ci) = mi, where ci = ⟨0, · · · , 0, 1R⟩ of length i+1; second, extend φ linearly to all

elements of F . That is, φ(∑

0≤i≤nrici) =

∑0≤i≤n

rimi. As φ is surjective,M ∼= F/ker(φ),

where ker(φ) = x ∈ F : φ(x) = 0.

Proposition 1.8 (RCA0) Let F =⊕i∈NR. For every surjective homomorphism

ε : B → C of R-modules and every homomorphism φ : F → C, there exists a

homomorphism ψ : F → B such that φ = εψ.

Proof: Let c0, c1, · · · be a basis of F . For each φ(ci) ∈ C, find an element di of B

such that ε(di) = φ(ci). Define for all i ∈ N, ψ(ci) = di.

Definition 1.21 (RCA0) Let P be a R-module. P is called projective if for every

surjective homomorphism ε : B → C of R-modules and every homomorphism φ :

P → C, there exists a homomorphism ψ : P → B such that φ = εψ.

The following proposition is an immediate consequence of Proposition 1.8.

Proposition 1.9 (Yamazaki [22])(RCA0) A free R-module is projective.

Theorem 1.19 (Yamazaki [22]) Over RCA0, a R-module is projective iff it is iso-

morphic to a direct summand of a free R-module.

22


A free R-module is projective, but the converse fails in general. However, clas-

sically, for a principal ideal domain R, projective R-modules are always free.

Definition 1.22 (RCA0) A commutative ring R with identity 1R is an integral

domain if it has no zero divisors, that is, for any a, b ∈ R− 0, ab = 0.

Downey, Lempp and Mileti in [23] showed that there is a computable integral

domain, say S, which is not a field such that S has no computable non-trivial proper

ideals which are finitely generated. In particular, S has no computable non-trivial

proper ideals which are principal. Hence, within RCA0, one cannot show that any

integral domain which is not a field has a principal non-trivial proper ideal. Indeed,

Downey, Lempp, and Mileti [23] proved that the statement that “every commutative

ring which is not a field has a finitely generated nontrivial proper ideal” is equivalent

to ACA0 over RCA0. This result of Downey, Lempp, and Mileti also implies that

“every commutative ring which is not a field has a principal non-trivial proper ideal”

is equivalent to ACA0 over RCA0, refer to Sato’s thesis [24] (Theorem 6.11).

In reverse mathematics, to prove basic properties on principal ideal domains

(PIDs) even within weak systems like RCA0, we had better to define a PID as an

integral ideal domain whose Σ01-ideals are generated by one element. Such PIDs are

known as Σ01-PIDs, refer to Simpson [25] and Sato [24] for more details.

Definition 1.23 (RCA0)

(1) Let R be a commutative ring with 1R. A sequence ⟨rn : n ∈ N⟩ ⊆ R is a

Σ01-ideal if for any i, j ∈ N, there is a k ∈ N such that rk = ri+ rj and for any

i ∈ N and s ∈ R, there is a j ∈ N such that rj = ris.

(2) An integral domain R is a Σ01-PID, if for any Σ0

1-ideal I of R, there is an

element r ∈ I such that I = (r), the ideal generated by r.

We will prove in Section 7.1 some results related to projective modules within

ACA0. In particular, we will show that ACA0 proves:

(1) every projective module over a Σ01-PID is free;

(2) every submodule of a projective module over a Σ01-PID is projective.

23


As a dual to projective modules, we have the following definition for injective

modules.

Definition 1.24 (RCA0) Let I be a R-module. I is injective if for any monomor-

phism µ : A→ B and any homomorphism φ : A→ I, there exists a homomorphism

ψ : B → I such that φ = ψµ.

The following classical lemma for injective modules is known as Baer’s criterion.

Lemma 1.3 (Baer’s Criterion) A module I over a commutative ring R with 1R is

injective iff for every ideal J of R and every R-module homomorphism f : J → I,

there is a homomorphism g : R → I such that g J= f .

Proposition 1.10 (RCA0) A module I over a commutative ring R with 1R is in-

jective implies for every ideal J of R and every R-module homomorphism f : J → I,

there is a homomorphism g : R → I such that g J= f .

Theorem 1.20 (Yamazaki [22]) The following are equivalent over RCA0.

(1) ACA0.

(2) If for every ideal J of R and every R-module homomorphism f : J → I, there

is a homomorphism g : R → I such that g J= f , then I is injective.

Definition 1.25 (RCA0) Let R be an integral domain. A R-module D is divisible

if for every d ∈ D and nonzero r ∈ R, there is a c ∈ D such that d = rc.

Classically, a module over a principal ideal domain is injective iff it is divisible.

Proposition 1.11 (RCA0) An injective module I over an integral domain R is

divisible.

Proof: For any nonzero r ∈ R and d ∈ I, let µ : R → R; 1R 7→ r and φ : R →I; 1R 7→ d. As µ(r′) = r′r = 0 implies r′ = 0, µ is a monomorphism. As I is an

injective R-module, there is a homomorphism ψ : R → I such that φ = ψµ. Now

d = φ(1R) = ψµ(1R) = ψ(r) = rψ(1R) with ψ(1R) ∈ I.

The following theorem about reverse mathematics of divisible abelian groups is

due to Friedman, Simpson and Smith [26] and can be found in Simpson’s book [20].

24


Theorem 1.21 [26] The following are equivalent over RCA0.

(1) ACA0.

(2) Every divisible abelian group is injective.

In Section 7.2, we will extend Theorem 1.21 to modules, and show that the

statement that “every divisible module over a Σ01-PID is injective” is equivalent to

ACA0 over RCA0. We will also show that ACA0 proves the statement that “every

quotient of an injective module over a Σ01-PID is injective”.

25

Part I

Bounded-jump operator and thehigh/low hierarchy

26

Chapter 2

A bounded-low c.e. set which islow, but not superlow

In this chapter, we will answer two questions raised by Anderson, Csima and Lange

in [2] by proving: (1) there is a bounded-low c.e. set which is low, but not superlow;

(2) 0′ contains a bounded-low c.e. set.

Anderson, Csima and Lange considered in [2] the interaction between the bounded-

jump operator and the Turing jump, and constructed a high bounded-low set. They

asked whether bounded-lowness and superlowness agree with each other on low c.e.

sets. We will give a negative answer to this question by showing the existence of

bounded-low c.e. sets which are low, but not superlow.

Theorem 2.1 There are low c.e. sets which are bounded-low but not superlow.

After seeing that superlow sets are always bounded-low, Anderson, Csima and

Lange [2] asked whether any superhigh set is also bounded-high. Our second result

in this chapter will provide a negative answer to this question. That is, we show the

existence of bounded-low c.e. sets which are of degree 0′. Thus, such bounded-low

c.e. sets are superhigh.

Theorem 2.2 0′ contains a bounded-low c.e. set.

We will give a proof of Theorem 2.1 first, and then prove Theorem 2.2 in Section

2.4.

To prove Theorem 2.1, we will construct a low c.e. set A such that A† is ω-c.e.

(so A is bounded-low), and A′ is not ω-c.e. (so A is not superlow). One approach

27

Chapter 2. A bounded-low c.e. set which is low, but not superlow

of constructing a set low, but not superlow, is to apply Sacks splitting theorem, to

split ∅′ into two low c.e. sets A and B. A and B cannot both be superlow, as if A is

superlow, then ∅′ is wtt-reducible to B [by a theorem of Bickford and Mills in [12]].

In same paper [12], Bickford and Mills also pointed out that there are superlow c.e.

sets A and B such that A⊕B computes ∅′. Most of the results in [12] can be found

in Nies’ book [27].

One feature of Sacks splitting is that there is no effective bound on the number of

injuries from a strategy with higher priority, and hence we cannot have a recursive

bound on the number of changes of A′(e) in advance. It also turns out to be an

obstacle for us to obtain a bounded-low c.e. set which is low, but not superlow, by

using Sacks splitting. Instead, we will construct a low, but not superlow, c.e. set

directly.

2.1 Requirements and strategies

We will construct a c.e. set A meeting the following requirements:

Le : If there are infinitely many stages s such that ΦAe (e)[s] ↓, then ΦA

e (e) ↓.

Re : The number of changes of A† at e is bounded by 2(e+ 1)2.

P⟨i,j⟩ : If φi and φ2j are both total, then there is some x such that either

A′(x) = limt→∞

φ2j(x, t)

or

|t ∈ N : φ2j(x, t) = φ2

j(x, t+ 1)| ≥ φi(x).

Here, φi : i ∈ N and φ2j : j ∈ N are standard enumerations of all partial

computable functions in one variable and two variables respectively, Φe : e ∈ N is

a standard enumeration of all partial computable functionals, and ⟨·, ·⟩ is an effective

bijection between N2 and N.

We assign the priority of requirements in our finite injury construction as

L0 ≺ R0 ≺ P0 ≺ L1 ≺ R1 ≺ P1 ≺ · · · ≺ Le ≺ Re ≺ Pe ≺ · · · .

28


If all the L-requirements are satisfied, then A is low. If all the R-requirements are

satisfied, then the whole construction will ensure that A† is ω-c.e., which ensures

that A is bounded-low. If all the P-requirements are satisfied, then A′ is not ω-c.e.,

and hence, A is not superlow.

The strategy of satisfying one Le is standard, i.e., we will set restraints to preservethe desired computations we have seen and this strategy can only be injured by P-

strategies with higher priority, which will happen at most finitely many times. Thus,

after a stage large enough, after which no P-strategies with higher priority act, if

the strategy sets restraints to preserve computations, these computations will be

preserved forever, and Le is satisfied.

The strategy for satisfying one Re is also to set restraints to preserve compu-

tations. At each stage s, when we see that φi(e) converges, where i ≤ e, we set a

restraint to protect A φi(e). Note that P-strategy with higher priority can injure

Re by enumerating small numbers x into A. But as φi(e) will be fixed, once Penumerates x into A, P selects a new number, x′ say, bigger than φi(e), and the

further enumeration of x′ will not injure Re [as this enumeration will not affect

the computation ΦAφi(e)e (e) if Φ

Aφi(e)e (e) converges]. Note that there are at most

e+1 many such restraints, and each P-strategy with higher priority can enumerate

numbers less than φi(e) at most once, the total number of such enumerations is at

most (e + 1)2. As one such an enumeration entails at most two changes of A†(e),

the number of changes of A†(e) is bounded by 2(e+ 1)2.

We now describe how to satisfy a P⟨i,j⟩-requirement. As in [27] for the construc-

tion of nonsuperlow sets, we will apply the recursion theorem for the construction.

That is, our construction will be uniform in parameters, r say, and in the r-th con-

struction, we will build a partial computable functional Γr, a c.e. set of axioms

⟨σ, n,m⟩, where if ⟨σ, n,m⟩ is enumerated into Γr at stage s, ΓAr (n)[s] is defined as

m with use As γAr (n)[s]= σ. From Γr, if Γr =Wr, we can have a computable function

pr such that

∀X ∀x[ ΓXr (x) ≃ ΦXpr(x)(pr(x)) ].

This will ensure that ΓAr (x) ↓⇐⇒ pr(x) ∈ A′. Hence, by controlling ΓAr , we can

defeat φ2j being an approximation of A′ with number of changes bounded by φi.

29


Since the construction is uniform in r, there is a computable function g such

that Γr = Wg(r). By recursion theorem, there is a r0 such that Wg(r0) = Wr0 , i.e.,

Γr0 = Wr0 . In the following, assume that we are in the r0-th construction, just fix

the computable function pr0 in advance.

A P⟨i,j⟩-strategy proceeds as follows:

(i) Choose a fresh witness x, let z = pr0(x).

(ii) Wait for φi(z) ↓ at some stage s.

(If we wait at (2) forever, then φi(z) ↑, and φi is partial and P⟨i,j⟩ is satisfied.)

(iii) Set N = φi(z) + 1, and run the following modules C(n) and D(n) for n ≤N . We start with C(0) in which case φ2

j(z, 0) ↓= 1 or D(0) in which case

φ2j(z, 0) ↓= 0, and let ℓ(−1) = 0. We will end when reaching C(N) or D(N),

because this shows that φi(z) cannot bound the number of changes of φ2j(z, q),

showing that P⟨i,j⟩ is satisfied.

C(n): There is a q1 > ℓ(n − 1) such that φ2j converges (at stage s) to 1 on all

(z, q) with ℓ(n−1) < q ≤ q1, and for those q2 with q1 < q2 ≤ s, if φ2j(z, q2)

converges at stage s, then φ2j(z, q2) = 1.

If such a q1 does not exist, then do nothing.

– If this is because of the fact that φ2j(z, ℓ(n− 1)+ 1) never converges,

then φ2j is partial and P⟨i,j⟩ is satisfied.

– If it is because of the existence of q2 with φ2j(z, q2) = 0, then wait for

a bigger stage at which φ2j(z, q) converges for all q ≤ q2, leave C(n)

with no actions and switch to D(n+ 1) with ℓ(n) = q2.

(Again, if φ2j(z, q) never converges for some q < q2 , then φ

2j is partial

and P⟨i,j⟩ is satisfied.)

That is, at stage s, we guess that φ2j(z, q) has limit 1, and we should not

keep such a guess if we see φ2j(z, q) converges to 0 after ℓ(n− 1).

Action for C(n): If ΓAr0(x) has definition at stage s− 1, then enumerate

γr0(x) into A, to undefine ΓAr0(x). Otherwise, do nothing.

30


– Wait for a stage t > s such that ΦAz (z)[t] diverges.

(For r0-th construction, such a stage t exists, because otherwise, we

would have ΓAr0(x) diverges, but ΦAz (z) converges.)

– Wait for a bigger stage at which we see that φ2j(z, q3) converges to

0, where q3 ≥ q1, and φ2j(z, q) converges for all q ≤ q3. Let ℓ(n) = q3

and switch to D(n+ 1).


2j is partial

and P⟨i,j⟩ is satisfied. Or, if φ2j(z, q) converges to 1 for all q ≥ q1,

then limx→∞

φ2j(z, q) = 1, while z ∈ A′, meaning that lim

x→∞φ2j(z, q) is not

the characteristic function of A′, and P⟨i,j⟩ is satisfied.)

D(n): There is a q1 > ℓ(n − 1) such that φ2j converges (at stage s) to 0 on all

(z, q) with ℓ(n−1) < q ≤ q1, and for those q2 with q1 < q2 ≤ s, if φ2j(z, q2)

converges at stage s, then φ2j(z, q2) = 0.

If there is no such a q1, then do nothing.

– If this is because of the fact that φ2j(z, ℓ(n− 1)+ 1) never converges,

then φ2j is partial and P⟨i,j⟩ is satisfied.

– If it is because of the existence of q2 with φ2j(z, q2) = 1, then wait

for a bigger stage at which φ2j(z, q) converges for all q ≤ q2, leave

D(n) with no actions and switch to C(n+1) with ℓ(n) = q2. (Again,

if φ2j(z, q) never converges for some q < q2 , then φ2

j is partial and

P⟨i,j⟩ is satisfied.)

That is, at stage s, we guess that φ2j(z, q) has limit 0, and we should not

keep such a guess if we see φ2j(z, q) converges to 1 after ℓ(n− 1).

Action for D(n): If ΓAr0(x) has no definition at stage s− 1, define ΓAr0(x)

with use γr0(x) big.

– Wait for a stage t > s such that ΦAz (z)[t] converges.

(For r0-th construction, such a stage t exists, because otherwise, we

would have ΓAr0(x) converges, but ΦAz (z) diverges.)

31


– Wait for a bigger stage at which we see that φ2j(z, q3) converges to

1, where q3 ≥ q1, and φ2j(z, q) converges for all q ≤ q3. Let ℓ(n) = q3

and switch to C(n+ 1).


2j is partial

and P⟨i,j⟩ is satisfied. Or, if φ2j(z, q) converges to 0 for all q ≥ q1,

then limx→∞

φ2j(z, q) = 0, while z ∈ A′, meaning that lim

x→∞φ2j(z, q) is not

the characteristic function of A′, and P⟨i,j⟩ is satisfied.)

Thus, a P⟨i,j⟩-strategy either waits at some C(n) or D(n) for some n < N , or

reaches C(N) or D(N), both of which will show that P⟨i,j⟩ is satisfied, as explained

above. Thus, in the construction, a P⟨i,j⟩-strategy can injure those strategies with

lower priority at most finitely often.

2.2 Construction

Construction of A.

Stage 0: Let A0 = ∅, and initialize all strategies.

Stage s > 0: Find a requirement with the highest priority, Q say, among

L0 ≺ R0 ≺ P0 ≺ · · · Ls ≺ Rs ≺ Ps

that requires attention at stage s, and act accordingly.

Say that Le requires attention at stage s if ΦAe (e)[s − 1] ↑ and ΦA

e (e)[s] ↓. Say

Re requires attention at stage s if there is a n ≤ e ≤ s such that φn(e) ↓ at stage s

and Re has not set A-restraint φn(e) yet.

For both cases, we act as follows:

Action: Set A-restraint as s, and initialize all strategies with lower priority.

Say that P⟨i,j⟩ requires attention at stage s if one of the following applies:

(i) P⟨i,j⟩ has no witness currently.

Action: Pick a fresh number x as a witness for P⟨i,j⟩.

32


(ii) x is selected, pr0(x) converges to z, φi(z)[s] ↓ and N is not defined yet.

Action: Definite N as φi(z) + 1.

(iii) N is defined, P⟨i,j⟩ is not in C(n) or D(n) for any n ≤ N , and φ2j(z, 0)[s]

converges.

Action: Let P⟨i,j⟩ start by running C(0), if φ2j(z, 0)[s] ↓= 1, or D(0), if

φ2j(z, 0)[s] ↓= 0. Here we let ℓ(−1) = 0.

(iv) P⟨i,j⟩ switches from C(n) to D(n+ 1), where n+ 1 < N .

Action: If ΓA(x) has no definition at stage s−1, then define ΓA(x)[s] = 0 with

use γ(x)[s] larger than all numbers used before.

(v) P⟨i,j⟩ switches from D(n) to C(n+ 1), where n+ 1 < N .

Action: If ΓA(x) has definition at stage s − 1, then enumerate γ(x) into A,

undefining ΓA(x).

(vi) P⟨i,j⟩ switches from C(N − 1) to D(N) or from D(N − 1) to C(N).

Action: Declare that φi(z) cannot bound the number of changes of φ2j(z, q),

and that P⟨i,j⟩ is satisfied.

If P⟨i,j⟩ acts as above, then all strategies with lower priority are initialized.

This ends the construction of A at stage s.

2.3 Verification

Lemma 2.1 For each requirement, Q say,

(1) Q can be initialized at most finitely often;

(2) Q acts at most finitely often and is satisfied;

(3) The restraint set by Q is finite, and Q can initialize all strategies with lower

priority finitely often.

33


Proof: We will show that for each e, (1)-(3) are true for all Le,Re,P⟨i,j⟩, where

⟨i, j⟩ = e. We prove it by induction.

When e = 0, L0 can never be initialized, due to its highest priority. (1) holds.

L0 acts and sets a restraint only when ΦA0 (0) converges, and once we set such a

restraint, ΦA0 (0) converges at any further stage, and (2) holds. In the construction,

L0 sets a restraint only when ΦA0 (0) converges, and after it acts, it will never act

again. (3) holds.

Note that R0 can be initialized by L0 at most once. (1) holds for R0. Note that

R0 acts only when φ0(0) converges, and after this, if ΦAφ0(0)0 (0) converges, then it

will converge forever, as no P can enumerate numbers less than φ0(0) into A. So

if ΦAφ0(0)0 (0) converges again, then it converges forever, which means that R0 is

satisfied. (2) holds. The restraint set by R0 is φ0(0), if it converges, and after it

acts, it will never act again. (3) holds.

For P⟨0,0⟩, it can be initialized by L0 and R0, so P⟨0,0⟩ can be initialized by L0

and R0 at most three times. (1) holds. Let s0 be the least stage after which P⟨0,0⟩

can not be initialized again, and suppose that P⟨0,0⟩ selects a witness x as a big

number at stage s1 > s0. Then x cannot be canceled later. Then after stage s1,

P⟨0,0⟩ acts only when

(i) it selects N ; or

(ii) it starts module C(0) or D(0); or

(iii) it switches from C(n− 1) to D(n), or from D(n− 1) to C(n); or

(iv) it reaches C(N) or D(N),

which can happen at most N+2 many times. If it reaches C(N) orD(N), then φ0(z)

cannot be a bound of the number of changes of φ20(z, q), and P⟨0,0⟩ is satisfied. If it

never selects N , then φ0(z) never converges (recall that pr0 is assumed to be total,

by the relativized smn -theorem). Here z = pr0(x). If it never starts module C(0) or

D(0), then φ20(z, 0) diverges. If it stays at some module C(n) or D(n) forever, then

either φ20(z, q) diverges for some q, or we will have lim

q→∞φ20(z, q) = A′(z). P⟨0,0⟩ is

satisfied for all cases. (2) holds for P⟨0,0⟩. Thus, after a stage s3 large enough, P⟨0,0⟩

34


will have no actions, and hence the restraint set by P⟨0,0⟩ is finite, and it will not

initialize strategies with lower priority. (3) holds.

For e > 0, we assume that the lemma is true for e′ < e. The proof that the

lemma is true for e is very similar to the proof for the basic case, i.e. e = 0, and

we will assume that after a stage s big enough, no strategies Le′ , Re′ , and Pe′ canact. Then we can show that after stage s, Le, Re, and Pe will behave exactly like

L0, R0, and P0 above, and hence the lemma is true for e.

This completes the proof of Lemma 2.1.

By Lemma 2.1, A is low but not superlow. Lemma 2.2 below shows that A is

bounded-low.

Lemma 2.2 A† is ω-c.e., and hence A is bounded-low.

Proof: To show that A† is ω-c.e., we need to show that there is a computable

function bounding the changes of e in A† for all e.

Fix e, and n ≤ e. Then in the construction, whenever we see φn(e) converges,

at stage s say, Re acts and initializes all strategies with lower priority. Thus, no

strategy with lower priority can put a number less that φn(e) into A later. So if

ΦAφn(e)e converges later (e enters A†), this computation can only be changed by Pe′-

strategies (where e′ < e, which have higher priority) by enumerating γAr0(y)[t] into

A. After this, if γAr0(y) is defined later, then it will be defined as a number bigger

than φn(e). Thus, because of this, ΦAφn(e)e is injured at most e + 1 many times,

then A†(e) can change at most 2(e+ 1) many times, for this particular n. Thus, in

total, A†(e) can change at most 2(e+ 1)2 many times, and A† is ω-c.e.

This completes the proof of Theorem 2.1.

2.4 A bounded-low set with Turing degree 0′

In this section, we give a sketchy proof of Theorem 2.2. That is, we will construct

a bounded-low c.e. set A Turing computing ∅′. We have seen how to construct

a bounded-low c.e. set in the proof of Theorem 2.1, so to prove Theorem 2.2, we

only need to show how to make A Turing complete. We will construct a partial

computable functional Γ such that ∅′ = ΓA. Of course, the construction of Γ will

be consistent with the R-strategies of making A bounded-low.

35


For the construction of Γ, we will make sure that for each e, ΓA(e) is defined

and computes ∅′(e) correctly. The idea of constructing Γ is quite standard. That

is, in the construction, at stage s, we find the least x with ΓA(x) not defined and

define ΓA(x)[s] = ∅′(x)[s] with use γ(x) big, like s. If later, at stage t > s, x enters

∅′, then at this stage, we enumerate γ(x) into A, undefining ΓA(x). In general, the

construction of Γ satisfies the following rules:

(i) For any x1, x2 with x1 < x2, if ΓA(x2) has definition at stage s, then ΓA(x1)

also has definition at this stage, with uses γ(x1) < γ(x2).

(ii) For any x1, x2 with x1 < x2, if ΓA(x1) is undefined at stage s, then ΓA(x2) is

also undefined at this stage, due to a number less than γ(x1) enters A.

(iii) For any x, if ΓA(x) has definition at stage s1 and s2, then γ(x)[s1] ≤ γ(x)[s2],

and if γ(x)[s1] < γ(x)[s2], ΓA(x) must have been undefined between these two

stages.

(iv) For any x, ΓA(x) can be undefined at most finitely often during the construc-

tion.

(v) For any x, ΓA(x) = ∅′(x).

We now show how to modify a Re-strategy in the proof of Theorem 2.1 so that it

is consistent with the rules above. Recall that the Re-strategy can act finitely many

times, and each time when it acts, because of φi(e) ↓ for some i ≤ e, it initializes

all strategies with lower priority. To make it consistent with the construction of Γ,

it needs to have an additional action: enumerate γ(e)[s] into A. This enumeration

undefines ΓA(e), so when it is defined again, the use will be defined as a big number,

which is bigger than the value φi(e). Thus, for a fixed e, the membership of e in

A† can be changed by Re′-strategies with e′ ≤ e, i.e., by further enumerations of

the γ-uses, or by rectifying ΓA(x) = 1 when some x < e enumerating into ∅′. Then

A†(e) changes at most 2(e+ 1)2 many times, which guarantees that A† is ω-c.e.

The construction and verification parts are standard finite injury arguments.

36

Chapter 3

A bounded-high set which is high,but not superhigh

Note that ∅′ is a bounded-high set which is also superhigh. In [2], Anderson, Csima

and Lange further asked whether there are bounded-high sets which are high but

not superhigh. Similarly, based on the existence of superhigh bounded-low sets, we

can also ask whether there are high but not superhigh bounded-low sets. In sections

3.1 and 3.2, we will provide two pseudo-jump inversion theorems for bounded-high

sets and bounded-low sets, which will provide positive answers to these questions.

Let We be a c.e. operator, the corresponding pseudo-jump operator Ve is defined

as V Ae = A ⊕WA

e for any set A. One way to obtain high, but not superhigh, c.e.

sets is through the pseudo-jump inversion theorem of Jockusch and Shore [28] which

says: for any c.e. operator We, there is a c.e. set C such that V Ce ≡T ∅′. This idea

was first used by Mohrherr in her paper [4].

The procedure to obtain high but not superhigh c.e. sets are:

(1) Through the construction of low but not superlow c.e. sets, one can obtain

a c.e. operator W such that for any set A, A ⊕WA is low over A, but not

superlow over A, i.e., (A⊕WA)′ ≤T A′, but (A⊕WA)′ tt A

′.

(2) By applying the pseudo-jump inversion theorem to this operator W , there

exists a c.e. set C such that C ⊕WC ≡T ∅′.

Then ∅′′ ≡1 (C ⊕WC)′ ≤T C′ (C is high), and ∅′′ ≡1 (C ⊕WC)′ tt C

′ (C is not

superhigh).

37

Chapter 3. A bounded-high set which is high, but not superhigh

In this chapter, we will provide two strengthened versions of Jockusch and Shore’s

pseudo-jump inversion theorem. That is, C in Jockusch and Shore’s theorem can

be bounded-high (section 3.1) or bounded-low (section 3.2).

Theorem 3.1 (1) For any c.e. operator W , there is a bounded-high set C such

that C ⊕WC ≡T ∅′.

(2) For any c.e. operator W , there is a bounded-low c.e. set C such that C ⊕WC ≡T ∅′.

The following two results follow Theorem 3.1 immediately:

Corollary 3.1 (1) There are bounded-high sets which are high but not superhigh.

(2) There are bounded-low c.e. sets which are high but not superhigh.

Proof: LetW be the c.e. operator such that for any set A, A⊕WA is low but not

superlow over A, then any C satisfying C ⊕WC ≡T ∅′ is high but not superhigh.

By Theorem 3.1, there are a bounded-high set C1 and a bounded-low c.e. set C2

such that C1 ⊕WC1 ≡T ∅′ and C2 ⊕WC2 ≡T ∅′. Thus C1 is a bounded-high set

which is high but not superhigh, while C2 is a bounded-low c.e. set which is high

but not superhigh.

We will prove Theorem 3.1.(1) in Section 3.1, and then prove Theorem 3.1.(2)

in Section 3.2.

For a c.e. operatorW , let Ψ be the corresponding partial computable functional.

That is, for any oracle A, e ∈ WA[s] ⇔ ΨA(e)[s] ↓. We view the use ψA(e)[s] of

ΨA(e)[s] as the use for e ∈ WA[s]. That is, if A ψA(e)[s] does not change later,

e will stay in WA. Ψ-uses satisfy the following usual use rules: if ΨA(e)[s] ↓ and

ΨA(e′)[s] ↓ with e < e′, then ψA(e)[s] < ψA(e′)[s]; if ΨA(e)[s] ↓, then ψA(e)[s] ≤ s;

and if ΨA(e)[s] ↑, then ψA(e)[s] = 0.

3.1 Pseudo-jump inversion via bounded-high sets

We will construct a ω-c.e. set C (so C ≤wtt ∅†) such that C ⊕ WC ≡T ∅′ and

∅†† ≤wtt C†. C will meet the following requirements:

38


Pe : ∅′(e) = ΓC⊕WC(e), where Γ is a computable functional built by us.

Qe : ∅††(e) = ∆C†g(e)(e), where ∆ is a computable functional built by us and g is

a computable function such that the use δC†(e) ≤ g(e) for all e.

Re : If there are infinitely many stages s with e ∈ WC [s], then e ∈ WC .

If all the R-requirements are satisfied, then WC is ∆02. By construction, C will

be ω-c.e., and hence C ⊕WC ≤T ∅′. If all the P-requirements are satisfied, then

∅′ ≤T C ⊕WC . Thus, ∅′ ≡T C ⊕WC . If all the Q-requirements are satisfied, then

∅†† ≤wtt C†, and C will be bounded-high.

We will apply the bounded-high strategy developed by Anderson, Csima and

Lange in [2], and combine it with Jockusch and Shore’s pseudo-jump inversion theo-

rem. Our construction is a finite injury argument, where the priority of requirements

is

R0 ≺ P0 ≺ Q0 ≺ R1 ≺ P1 ≺ Q1 ≺ · · ·Re ≺ Pe ≺ Qe ≺ · · · .

3.1.1 A Re-strategy

The original Re-strategy is just setting a C-restraint ψC(e)[s] when e ∈ WC [s] −WC [s− 1] (i.e., ΨC(e)[s] ↓ and ΨC(e)[s− 1] ↑).

We first define C-restraint

r(e, s) = maxψC(e)[s′] : s′ ≤ s ∧ΨC(e)[s′] ↓.

Then r(e, s) ≤ r(e, s+ 1). Suppose that e ∈ WC [s]−WC [s− 1]. e will be removed

out of WC [t] at some least stage t > s only if

Ct ψC(e)[s] = Cs ψC(e)[s] .

So if e ∈ WC [s] and Cs r(e,s)= Ct r(e,s) for all t ≥ s, then e ∈ WC [t] for all stages

t ≥ s, and thus e ∈ WC .

We say that Re is injured at stage s if either some Pi enumerates the Γ-use

γC⊕WC(i)[s] ≤ r(e, s) into C when i ∈ ∅′[s]−∅′[s− 1]; or some Qi enumerates its

coding indicator c ≤ r(e, s) into C or extracts its coding indicator c ≤ r(e, s) out of

C when redefining ∆C†g(i)(i)[s] = ∅††[s](i) (the definition of coding indicators will

be clarified in the following basic Q-strategy).

39


During the construction, we do not allow positive requirements with lower prior-

ity, namely Pj,Qj with j ≥ e, to enumerate elements ≤ r(e, s) into C or to extract

elements ≤ r(e, s) out of C, at any stage s (this will be clear in Case 1 for Pj andCase 2 for Qj), and hence these requirements cannot injure Re later.

Assume that e ∈ WC [s] − WC [s − 1] with C-restraint r(e, s). Since we have

appointed a C-restraint r(e, s − 1), we only need to consider the case r(e, s) >

r(e, s− 1). So we assume that at stage s, r(e, s) > r(e, s− 1).

Case 1. At stage s with r(e, s) > r(e, s − 1), for Pj with j ≥ e, we will show

that ΓC⊕WC(e)[s] ↑. That is, for all stages s′ < s at which ΓC⊕WC

(e)[s′] ↓, either

(a) : WC [s] γC⊕WC (e)[s′] =WC [s′] γC⊕WC (e)[s′]

or

(b) : Cs γC⊕WC (e)[s′] = Cs′ γC⊕WC (e)[s′]

holds.

Indeed, for stages s′ < s, if ΓC⊕WC(e)[s′] ↓ with ΨC(e)[s′] ↑, then e /∈ WC [s′].

As e ∈ WC [s],

WC [s] e =WC [s′] e

(recall that the symbol X z denotes the set x ∈ X : x ≤ z). As we will choose

γC⊕WC(e)[s′] > e, (a) holds.

If ΓC⊕WC(e)[s′] ↓ with ΨC(e)[s′] ↓. By the assumption that e ∈ WC [s]−WC [s−1]

with new computation ΨC(e)[s] ↓, we have

Cs ψC(e)[s′] = Cs′ ψC(e)[s′] .

Claim γC⊕WC(e)[s′] > ψC(e)[s′], then (b) holds.

We now prove

γC⊕WC

(e)[s′] > ψC(e)[s′]

for all s′ < s with ΓC⊕WC(e)[s′] ↓ and ΨC(e)[s′] ↓. Suppose that the computation

ΓC⊕WC(e)[s′] is first defined at stage t′ ≤ s′ and returns back at stage s′. This means

that

Cs′ γC⊕WC (e)[t′]= Ct′ γC⊕WC (e)[t′] ∧ WC [s′] γC⊕WC (e)[t′]= WC [t′] γC⊕WC (e)[t′] .

There are two cases:

40


(1) ΨC(e)[t′] ↓. As γC⊕WC(e)[t′] is defined to be bigger than all used numbers,

γC⊕WC

(e)[t′] > ψC(e)[t′].

Then ΨC(e)[t′] also comes back at stage s′, we have

γC⊕WC

(e)[s′] = γC⊕WC

(e)[t′] > ψC(e)[t′] = ψC(e)[s′].

(2) ΨC(e)[t′] ↑. Then e /∈ WC [t′], but by assumption, e ∈ WC [s′]. As γC⊕WC(e)[t′] >

e,

WC [t′] γC⊕WC(e)[t′] = WC [s′] γC⊕WC

(e)[t′] .

This contradicts with the assumption that the old computation ΓC⊕WC(e)[t′] ↓

returns back at stage s′, so this case is indeed impossible.

In construction, at each stage t, if ΓC⊕WC(e)[t] ↓ and ΓC⊕WC

(j)[t] ↓ with e < j,

we will make the corresponding uses satisfying γC⊕WC(e)[t] < γC⊕WC

(j)[t]. So when

ΓC⊕WC(e)[s] is undefined, then all ΓC⊕WC

(j) with j > e are also undefined at stage

s.

Hence, when the C-restraint r(e, s) moves to a higher level, all the previous

ΓC⊕WC(j)-computations with j ≥ e are undefined, we will define new computations

ΓC⊕WC(j) with use > r(e, s). Then the enumerations of such uses into C can not

injure Re as long as r(e, s) does not increase later.

Case 2. At stage s with r(e, s) > r(e, s−1), for Qj with j ≥ e, we first consider

Qe. Say that Qe is injured at stage s if there is an i ≤ e such that the C-restraint

r(i, s) on Ri is larger than or equal to the coding indicator of Qe.

The strategy of Qe to deal with this injury from Ri is still to undefine previ-

ously defined ∆C†g(e)(e)-computations and then define ∆C†g(e)(e) via new coding

indicators > r(i, s). Then Qe will not injure Ri as long as r(i, s) does not increase

later.

To undefine ∆C†g(e)(e)-computations, we need to change C† below g(e). In

order to enumerate elements into or extract elements out of C†, we will build a

partial computable function q and a partial computable functional Θ, and assume

that the recursion theorem provides us the graph of q and the c.e. relation of Θ.

41


Then by controlling q and Θ, we can move elements (i.e., indices of Θ obtained

by smn -theorem) into or out of C† (this will be much clear in the following basic

Qe-strategy).

Hence, during the construction, Re can only be injured by positive requirements

with higher priority. Each Pi(i < e) can injure Re at most once, and each Qi(i < e)

can injure Re finitely often by enumerating coding indicators. So Re can be injured

finitely often, and thus limsr(e, s) <∞.

3.1.2 A Pe-strategy

We will build a functional Γ such that ∅′ = ΓC⊕WC. At each stage s, when e ∈

∅′[s+ 1]−∅′[s] and ΓC⊕WC(e)[s] ↓, we will enumerate the use γC⊕WC

(e)[s] into C

to undefine ΓC⊕WC(e), and then redefine ΓC⊕WC

(e) = 1 with big use.

When the C-restraints r(i, s + 1) on Ri with i ≤ e move up, i.e. r(i, s + 1) >

r(i, s), as in a basic Ri-strategy, we see ΓC⊕WC(i) is undefined at the beginning of

stage s+1. Thus, ΓC⊕WC(j) ↑ for all j ≥ i, and such ΓC⊕WC

(j) will be defined later

with new use > r(i, s+ 1).

During the construction, we will define at most one ΓC⊕WC(y) at each stage. At

the last step of stage s+ 1, we find the least y with ΓC⊕WC(y) ↑ currently.

If ΓC⊕WC(y) is never defined before, or there is a largest stage w < s+1 at which

ΓC⊕WC(y)[w] ↓ and also at least one of the following three conditions holds:

(1) y ∈ ∅′[w + 1]−∅′[w];

(2) y ∈ WC [t+ 1]−WC [t] with r(y, t+ 1) > r(y, t) for some t ∈ [w, s];

(3) γC⊕WC(y)[w] ≤ γC⊕WC

(y − 1)[s].

Then define ΓC⊕WC(y)[s + 1] = ∅′[s + 1](y) with use γC⊕WC

(y)[s + 1] bigger than

all numbers used before.

Otherwise, there is a largest stage w < s+ 1 at which ΓC⊕WC(y)[w] ↓ and none

of three conditions above holds, just define ΓC⊕WC(y)[s+1] = ∅′[s+1](y) with use

γC⊕WC(y)[s+ 1] = γC⊕WC

(y)[w].

We now show that limsγC⊕WC

(e)[s] <∞. Assume that limsγC⊕WC

(e−1)[s] <∞,

and that limsr(i, s) <∞ for all i ≤ e. We can find a stage s0 such that for all i ≤ e,

42


• limsr(i, s) = r(i, s0),

• i ∈ ∅′ ⇔ i ∈ ∅′[s0].

Let s1 ≥ s0 + 1 be the least stage such that ΓC⊕WC(e)[s1] ↓. Then after stage s1,

γC⊕WC(e)[s] will not move up, so lim

sγC⊕WC

(e)[s] = γC⊕WC(e)[s1].

3.1.3 A Qe-strategy

Fix the canonical notation for ω2. That is, for each β = ω · i+ j, we view β coded as

⟨i, j⟩, and the corresponding order on the set of all ordered pairs of natural numbers

is just the lexicographic order, i.e., ⟨i, j⟩ < ⟨k, l⟩ iff i < k or i = k with j < l.

Anderson and Csima proved in [1] that ∅†† is ω2-c.e. under this notation, i.e., there

is a partial computable function χ : ω × ω2 → 0, 1 such that

• for each e, there is a least ordinal βe < ω2 such that χ(e, βe) ↓= ∅††(e).

Then

• for each e, there is a first stage se and a least ordinal βe,se = ω · ise + jse such

that χ(e, βe,se)[se] ↓;

• for each e and stage s ≥ se, there is a least ordinal βe,s = ω · ie,s + je,s such

that χ(e, βe,s)[s] ↓.

Now βe = limsβe,s, and ∅††(e) = χ(e, βe) = lim

sχ(e, βe,s). As χ is partial computable,

ise and jse are computable functions on e; ie,s and je,s are computable functions on

e, s.

For convenience, we can also choose χ such that

• for each e and stage s ≥ se, if βe,s = βe,s+1, then χ(e, βe,s+1) = χ(e, βe,s).

In this case, βe,s+1 < βe,s, that is, either ie,s+1 < ie,s, or ie,s+1 = ie,s with

je,s+1 < je,s.

43


Fix a partial computable function χ : ω× ω2 → 0, 1 which satisfies conditions

above. At stage s ≥ se, we view ∅††[s](e) = χ(e, βe,s).

Consider the requirement Qe : ∅††(e) = ∆C†g(e)(e). At stage s′ ≥ se, we define

∆C†g(e)(e)[s′] = χ(e, βe,s′). If there is a least stage s > s′ at which βe,s < βe,s−1 or Qe

is injured by R-strategies with higher priority (i.e., the current coding indicator of

Qe is less than or equal to the C-restraint on R), then we need to rectify functional

∆C†g(e)(e) at stage s by enumerating some x ≤ g(e) into C† or extracting some

y ≤ g(e) out of C†.

In order to enumerate elements into or extract elements out of C†, we will build

additional partial computable functions and partial computable functionals. By us-

ing the recursion theorem, we can change C† on indices of such partial computable

functionals. The construction indeed relies on parameters, say r. In the r-th con-

struction, we will build an auxiliary partial computable function qr, an auxiliary

partial computable functional Θr, and also an auxiliary c.e. set Qr, where Qr en-

codes the graph of qr as well as the c.e. relation of Θr.

Assume that we are in the r-th construction. To encode ∅††(e) into ∆C†(e), we

will reserve three types of markers of Qe, namely injury markers, location markers

and coding markers [such markers will be defined in Case 1 and Case 2 below].

We build the functional ∆ such that ∆C†(e) only depends on the membership of

such markers in C†. Let Ue,r be the set of all injury markers, location markers

and coding markers defined during the r-th construction. Then the use function

δC†(e) = maxUe,r. We will determine a strictly increasing computable function gr

(in section 3.1.4) such that gr(e) ≥ maxUe,r. Then ∅††(e) = ∆C†gr(e)(e).

Case 1. We first consider the situation when Qe is injured from negative re-

quirements with higher priority.

Injury markers. When the C-restraint r(i, s) on Ri(i ≤ e) increases to a new

level at stage s with r(i, s) ≥ c, the coding indicator of Qe [the definition of cod-

ing indicators will be given in the following Case 2.1], we will enumerate an injury

marker < gr(e) into C† and never remove it out. This enumeration injures all old

convergence ∆C†gr(e)(e)-computations, and ∆C†gr(e)(e) will arrive at new computa-

tions. In this case, just cancel existed coding indicators and coding markers of Qj

with j ≥ e.

44


In order to define injury markers, we first choose a fresh witness, x say. Now

qr(x) and ΘYr (x) (for any oracle Y ) are undefined currently. Define qr(x) = 0 and

ΘYr (x) = 0 for all oracles Y at stage s, and then enumerate ⟨⟨x, 0⟩, ⟨λ, x, 0⟩⟩ into

Qr,s (note we also enumerate other elements of the form ⟨⟨y, c⟩, ⟨σ, y, 0⟩⟩ with σ a

binary string into Qr when defining location markers and coding markers), where

⟨·, ·⟩ : N2 → N and ⟨·, ·, ·⟩ : N3 → N are fixed effective one-to-one functions; λ stands

for the empty string [we indeed fix an effective coding of 2<ω (the set of binary

strings) into N and view a binary string σ in ⟨σ, y, 0⟩ as its code].

We now provide more details for the use of recursion theorem.

The application of recursion theorem.

First, by smn -theorem, there are computable functions f1 and f2 such that for

each x, Wf1(x) = z : ∃y[⟨z, y⟩ ∈ Wx], and Wf2(x) = y : ∃z[⟨z, y⟩ ∈ Wx], whereWx is the x-th c.e. set of natural numbers.

In the r-th construction, if Qr =Wr, then

Wf1(r) = z : ∃y[⟨z, y⟩ ∈ Qr]

and

Wf2(r) = y : ∃z[⟨z, y⟩ ∈ Qr].

So the graph of qr is Wf1(r), and the c.e. relation of Θr is just Wf2(r). Then again,

we can use smn -theorem as well as the relativized smn -theorem to obtain the injective

computable functions kr and hr such that for n,m and Y ,

φkr(n)(m) = qr(n), ΦYhr(n)(m) ≃ ΘY

r (n).

Moreover, by Padding Lemma, we can further choose kr and hr such that kr(n) <

hr(n) for all n. Such kr and hr will be used to define injury markers, location

markers as well as coding markers.

However, the assumption that Qr = Wr may be incorrect. Then our guess

on qr and Θr are wrong, and hence kr and hr are also incorrect. Thus, the r-th

construction does not work in general.

Observe that the c.e. sets Qr(r ∈ N) are uniform in r, then Qr = Wf(r) for

some computable function f . By recursion theorem, there is at least one r0 (indeed

45


infinitely many) such that Qr0 =Wf(r0) =Wr0 . Then in the r0-th construction, our

guess on qr0 and Θr0 are both correct, and thus, the injective functions kr0 and hr0

really satisfy for all n,m and Y ,

φkr0 (n)(m) = qr0(n), ΦYhr0(n)

(m) ≃ ΘYr0(n).

Then Qe can be satisfied via the r0-th construction.

This ends the description of applying recursion theorem.

Now return back to the fresh witness x above which devotes to define injury

markers. At stage s, after defining qr(x) = 0 and ΘYr (x) = 0 for all oracles Y , we

view Φ∅hr(x)

(hr(x)) ↓= Θ∅r (hr(x)) ↓= 0, and for all m, φkr(x)(m) ↓= qr(x) ↓= 0.

Then

ΦCφkr(x)

(hr(x))

hr(x)(hr(x)) ↓ ∧ kr(x) < hr(x).

This means that hr(x) enumerates into C† forever. Call hr(x) an injury marker of

Qe.

As we will determine an increasing computable function gr with gr(e) bigger than

all injury markers of Qe, ∆C†gr(j)(j) ↑ for all j ≥ e. We can omit such Qj’s coding

indicators and coding markers, and Qj will obtain new coding indicators larger than

the C-restraint r(i, s).

Case 2. We now consider the situation when βe,s < βe,s−1. There are two

subcases.

Case 2.1. If the approximation of ∅††(e) changes at stage s via ie,s < ie,s−1, then

we will undefine all old definitions and then define ∆C†gr(e)(e) via new computations.

Location markers. To undefine old computations, we will enumerate a new num-

ber < gr(e) into C† and never remove it out. Such a number is called a location

marker at level ie,s, which indicates that approximations of ∅††(e) move to new level

ie,s and that ∆C†gr(e)(e) will be defined at that new level.

To define location markers, we choose a big number, say y, and then define

qr(y) = 0, ΘYr (y) = 0 for all oracles Y , and enumerate ⟨⟨y, 0⟩, ⟨λ, y, 0⟩⟩ into Qr,s.

Then we view

ΦCφkr(y)

(hr(y))

hr(y)(hr(y)) ↓ ∧ kr(y) < hr(y);

46


by φkr(y)(hr(y)) = 0, hr(y) ∈ C†[t] for all t ≥ s. Say that hr(y) is a location marker

at level ie,s. Now ∆C†gr(e)(e) is undefined, because we will define gr(e) bigger than

all location markers of Qe.

Coding markers and coding indicators. After obtaining location markers above,

we immediately define coding markers by picking a new number, say c, and then

define a coding marker via c as follows. Again, choose a number z larger than all

used numbers. Define

qr(z) = c

and

ΘYr (z) :=

0, if c /∈ Y↑, otherwise.

We enumerate all elements of the form ⟨⟨z, c⟩, ⟨σ, z, 0⟩⟩ with σ ∈ 2<ω, |σ| = c + 1

and σ(c) = 0 into Qr,s, where |σ| is the length of σ.

We think for all m,

φkr(z)(m) ↓= qr(z) ↓= c, ΦYhr(z)(m) ≃ ΘY

r (z).

So for all stages t ≥ s,

ΦCchr(z)

(hr(z))[t] ↓⇔ ΘCcr (z)[t] ↓⇔ c /∈ Ct.

Then we view hr(z) ∈ C†[t] ⇔ c /∈ Ct. Such a hr(z) is called a coding marker at

level ie,s, and such a witness c is called the coding indicator of hr(z).

We will define ∆C†gr(e)(e) via the coding marker hr(z) and the coding indicator

c at level ie,s. If χ(e, βe,s) = 1, then keep c /∈ Cs and define

∆C†gr(e)(e)[s] = C†[s](hr(z)) = 1.

Otherwise, χ(e, βe,s) = 0, to undefine ΘCsr (z), just enumerate the coding indicator

c into Cs. Then ΦCs

hr(z)(m) ↑ for all m; in particular, ΦCs

hr(z)(hr(z)) ↑, and thus

hr(z) /∈ C†[s]. Define

∆C†gr(e)(e)[s] = C†[s](hr(z)) = 0.

47


Case 2.2. If ie,s = ie,s−1, but the approximation χ(e, βe,s) of ∅††(e) changes at

stage s via je,s < je,s−1. As the approximation stays in the previous level, we have

defined the coding indicator c and the coding marker hr(z) at level ie,s, and defined

∆C†gr(e)(e) via c and hr(z). At stage s− 1, we have

∆C†gr(e)(e)[s− 1] = χ(e, βe,s−1) = C†[s− 1](hr(z)) = 1− Cs−1(c).

If χ(e, βe,s) = 0, then χ(e, βe,s−1) = 1, so c /∈ Cs−1. Just enumerate c into Cs.

If our construction is the r0-th construction provided by recursion theorem, this

enumeration undefines ΦCs

hr0 (z)(hr0(z)), so hr0(z) /∈ C†[s]. We can define

∆C†gr0 (e)(e)[s] = 0 = χ(e, βe,s) = C†[s](hr0(z)) = 1− Cs(c).

If χ(e, βe,s) = 1, then χ(e, βe,s−1) = 0, so c ∈ Cs−1. Now extract c out of

C. Similarly, if we are in the r0-th construction, then ΦCchr0 (z)

(hr0(z))[s] ↓ with

φkr0 (z)(hr0(z)) = c and kr0(z) < hr0(z), and hence hr0(z) ∈ C†[s]. We can define

∆C†gr0 (e)(e)[s] = 1 = χ(e, βe,s) = C†[s](hr0(z)) = 1− Cs(c).

3.1.4 Define the computable function gr

We now specify a computable function gr in the r-th construction such that for all

e, gr(e) is bigger than maxUe,r, where Ue,r is the set of injury markers, location

markers and coding markers of Qe defined in the r-th construction.

We first determine a recursive upper bound for the size |Ue,r| of Ue,r. For injurymarkers, we define them only if someRi(i ≤ e) has its C-restraint r(i, s) increase. So

we also determine a number µ(i) which exceeds the number of times r(i, s) increases.

As R0 has the highest priority, when 0 is enumerated into WC [s] for first time,

we set C-restraint r(0, s) = ψC(0)[s] and preserve it forever. Thus, r(0, s) increases

at most once, and let µ(0) = 1.

Now consider |U0,r|. Q0 is injured at most once when r(0, s) increases, so it

has at most one injury marker. For location markers, we define them when the

approximation of ∅††(0) decreases to a new level. Recall that s0 is the first stage at

which there is a least ordinal β0,s0 = ω · is0 + js0 such that χ(0, β0,s0)[s0] ↓, so there

are at most is0 + 1 many levels for Q0. Thus, the number of location markers of

48


Q0 is at most is0 . For coding markers, we change them when Q0 is injured or the

approximation of ∅††(0) arrives at a new level, then the number of coding markers

of Q0 (= the number of coding indicators of Q0) is at most µ(0) + is0 + 1. Hence,

|U0,r| ≤ µ(0) + is0 + (µ(0) + is0 + 1).

In order to find an upper bound for general |Ue,r|, we also determine a number

ν(e) which bounds the number of coding markers of Qe. Set ν(0) = µ(0) + is0 + 1.

Assume that we have already obtained µ(e′) and ν(e′) for all e′ < e.

Now determine µ(e) which bounds the number of times C-restraints r(e, s) on

Re increases. Re is injured at stage s if some Pi(i < e) enumerates Γ-uses ≤ r(e, s)

into C, or some Qi(i < e) enumerates coding indicators ≤ r(e, s) into C or removes

coding indicators ≤ r(e, s) out of C. Note that each Pi(i < e) will enumerate at

most one use into C. Re will be injured by P-requirements at most 2e many times.

For a sequence of fixed coding indicators, say ci, of Qi with i < e, the number of

configurations of C at this sequence is 2e, and there are at most ν(i) many coding

indicators of Qi(i < e). Re will be injured by Q-requirements for at most 2e∏

0≤i<eν(i)

many times.

In total, Re will be injured by positive requirements at most 22e∏

0≤i<eν(i) many

times. Moreover, r(e, s) can move up before or after each injury, hence, r(e, s) will

increase at most 22e+1∏

0≤i<eν(i) times. Just set µ(e) = 22e+1

∏0≤i<e

ν(i).

Now we can define ν(e) which bounds the number of coding markers of Qe. We

define coding markers for Qe when it is injured by Ri(i ≤ e) in which case r(i, s)

increases, or the approximation of ∅††(e) arrives at a new level. Recall that se is the

first stage at which there is a least ordinal βe,se = ω·ise+jse such that χ(e, βe,se)[se] ↓.Qe has at most ise + 1 many levels. Then we can set ν(e) = ise + 1 +

∑0≤i≤e

µ(i).

Now we are ready to determine a recursive upper bound for |Ue,r|. Qe has at

most ise many location markers; and Qe is injured byRi with i ≤ e at most∑

0≤i≤eµ(i)

many times, so it has at most∑

0≤i≤eµ(i) many injury markers; and Qe has at most

ν(e) many coding markers. Thus in total,

|Ue,r| ≤ ise +∑0≤i≤e

µ(i) + ν(e) ≤ 2ν(e).

49


This ends the description of bounding the size of Ue,r.

Our initial target is to bound maxUe,r, the use of ∆C†(e). In order to recur-

sively bound maxUe,r, during the r-th construction, we will choose Γ-uses for P-

requirements and the witnesses at which we define qr and Θr for Q-requirements in

a neat way. Fix an effective injection ⟨, ⟩ : N × N → N such that for all i, j ∈ N,⟨i, j⟩ is strictly bigger than both i and j, and if i < k and j < l, then ⟨i, j⟩ is strictlyless than ⟨k, l⟩. Let ⟨i,N⟩ := ⟨i, j⟩ : j ∈ N.

All Γ(e)-uses for Pe are chosen from ⟨3e,N⟩. Γ(e)-uses will be moved up only if

some i ≤ e enumerates into ∅′, or the C-restraints on some Ri with i ≤ e increases.

The witnesses for defining injury markers, location markers, and coding markers

of Qe are chosen from ⟨3e+1,N⟩, and there are at most 2ν(e) many such witnesses.

Then during the construction, we choose such witnesses of Q0 from

⟨1, x⟩ : x ≤ 2ν(0)

one by one in increasing order, and choose such witnesses of Qe from

⟨3e+ 1, x⟩ : e+ 2∑

0≤i≤e−1

ν(i) ≤ x ≤ e+ 2∑0≤i≤e

ν(i)

one by one in increasing order.

The coding indicators for coding markers of Qe are chosen from ⟨3e + 2,N⟩,and there are at most ν(e) many such coding indicators. During the construction,

when the C-restraint on some Ri(i ≤ e) exceeds the coding indicators of Qe or the

approximations of ∅††(e) decrease into a new level, we omit old coding indicators

of Qe, and appoint new coding indicators form ⟨3e+ 2,N⟩.During the r-th construction:

(1) When defining injury markers or location markers for Qe, we first pick a least

unused witness, say ⟨3e+ 1, y⟩, with y satisfying:

– e+ 2∑

0≤i≤e−1

ν(i) ≤ y ≤ e+ 2∑

0≤i≤eν(i),

– qr(⟨3e+ 1, y⟩) ↑,

50


– ΘYr (⟨3e+ 1, y⟩) ↑.

By |Ue,r| ≤ 2ν(e), during the r-th construction, we will define qr and ΘYr on

no more than 2ν(e) many arguments inside ⟨3e+1,N⟩, so such a y exists. We

then define

qr(⟨3e+ 1, y⟩) = 0

and

ΘYr (⟨3e+ 1, y⟩) = 0

for all oracles Y , and view

Φ∅hr(⟨3e+1,y⟩)(hr(⟨3e+ 1, y⟩)) ↓= 0 ∧ φqr(⟨3e+1,y⟩) ↓= 0

with qr(⟨3e+1, y⟩) < hr(⟨3e+1, y⟩). That is, the injury marker or the location

marker hr(⟨3e+ 1, y⟩) is enumerated into C† forever.

(2) When defining coding markers of Qe, we first choose a new coding indicator,

say ⟨3e + 2, x⟩, with x bigger than all numbers used before, and then pick a

least unused witness, say ⟨3e+ 1, z⟩, with z satisfying:

– e+ 2∑

0≤i≤e−1

ν(i) ≤ z ≤ e+ 2∑

0≤i≤eν(i),

– qr(⟨3e+ 1, z⟩) ↑,

– ΘYr (⟨3e+ 1, z⟩) ↑.

Similarly, such a z exists. Then we define

qr(⟨3e+ 1, z⟩) = ⟨3e+ 2, x⟩,

and for all oracles Y , define

ΘYr (⟨3e+ 1, z⟩) ↓⇔ ⟨3e+ 2, x⟩ /∈ Y.

We view the coding marker hr(⟨3e+1, z⟩) is enumerated into C† iff its coding

indicator ⟨3e+ 2, x⟩ is outside C.

51


Hence, all markers of Qe are of the form hr(⟨3e+ 1, y⟩) for some

e+ 2∑

0≤i≤e−1

ν(i) ≤ y ≤ e+ 2∑0≤i≤e

ν(i).

As hr is a computable function, we can define

gr(0) =∑

0≤y≤2ν(0)

hr(⟨1, y⟩).

For e > 0, define gr(e) by induction as

gr(e) = 1 + gr(e− 1) +∑

e+2∑

0≤i≤e−1

ν(i)≤ y ≤ e+2∑

0≤i≤e

ν(i)

hr(⟨3e+ 1, y⟩).

Then gr is a strictly increasing computable function such that for all e, maxUe,r <

gr(e).

In the r-th construction, fix such a gr first, and for each e, we try to ensure

∅††(e) = ∆C†gr(e)(e) with the actual use δC†(e) = maxUe,r.

In the following formal construction and verification, assume that we are in

the r0-th construction, which is provided by recursion theorem. Then the injective

computable functions kr0 , hr0 , gr0 are all correct, we denote them respectively as

k, h, g for simplicity.

3.1.5 Construction

Recall that se is the first stage at which there is an ordinal α < ω2 such that

χ(e, α)[se] ↓. Assume se ≥ e, and let βe,se = ω · ise + jse be the least such an α. At

stage s ≥ se, let βe,s = ω · ie,s+ je,s be the least ordinal such that χ(e, βe,s)[s] ↓, and

we assume ie,s, je,s ≤ s.

Assume that there is at most one element ≤ s enumerating into ∅′ at each stage

s.

Construction of C.

Let C0 := ∅, and all functionals are totally undefined at stage 0.

52


Stage s+ 1:

(1) Coding ∅′. If there is an i ≤ s + 1 such that i ∈ ∅′[s + 1] − ∅′[s] with

ΓC⊕WC(i)[s] ↓, then put γC⊕WC

(i)[s] into C. Thus, ΓC⊕WC(i) is undefined.

(2) Coding ∅††. Say Qe requires attention at stage s + 1 if s+ 1 ≥ se and one of

the following conditions holds.

(Q0) Qe has no coding indicators at the end of stage s.

(Q1) (Q0) fails and Qe has an uncanceled coding indicator c, and some Ri

with i ≤ e has its C-restraint r(i, s) ≥ c.

(Q2) (Q0) and (Q1) both fail, and βe,s+1 < βe,s with s + 1 > se, in this case

χ(e, βe,s+1) = χ(e, βe,s).

Find all Qe(e ≤ s + 1) which require attention at stage s + 1, and consider

them in order of priority.

Now assume that we arrive at Qe which requires attention.

(2.1) When (Q0) holds.

– If there are no location markers at level ie,s+1, first define a location

marker at level ie,s+1 as in (⋆) of (2.3.1).

– Define coding indicator and coding marker as in (⋆⋆) of (2.3.1).

– Define computation ∆C†g(e)(e) via the new coding indicator and coding

marker as in (⋆ ⋆ ⋆) of (2.3.1).

Check whether there is a requirement Qj(e < j ≤ s + 1) which requires

attention at stage s + 1. If there is such a Qj, find the highest priority one,

act correspondingly. Otherwise, go to step (3).

53


(2.2) When (Q1) holds.

First define an injury marker to undefine all old convergence computations of

∆C†g(e)(e) as follows:

– pick the least unused number z in

⟨3e+ 1, y⟩ : e+ 2∑

0≤i≤e−1

ν(i) ≤ y ≤ e+ 2∑0≤i≤e

ν(i);

– define qr0(z) = 0 and ΘYr0(z) = 0 for all oracles Y .

By recursion theorem, for all m, φk(z)(m) ↓= 0, Φ∅h(z)(m) ↓= 0, we also choose

k(z) < h(z). So h(z) ∈ C†[t] for all t ≥ s + 1. h(z) is the desired injury

marker.

As we choose g such that h(z) < g(e), ∆C†g(e)(e) ↑, thus all ∆C†g(j)(j) ↑ with

j ≥ e. Say Qj(j ≥ e) is injured at stage s + 1. Cancel all existed coding

indicators and coding markers of Qj(j ≥ e), go to step (3).

(2.3) When (Q2) holds, βe,s+1 < βe,s. There are two subcases (2.3.1) and (2.3.2).

(2.3.1) ie,s+1 < ie,s, the approximation of ∅††(e) now moves to a new level. Cancel

the coding indicator and coding marker at level ie,s. Define a location marker

at level ie,s+1 to undefine all old definitions of ∆C†g(e)(e), and then define a

coding marker to encode ∅††[s+ 1](e), which is χ(e, βe,s+1).

(⋆) Define the location marker. Pick the least unused number z in

⟨3e+ 1, y⟩ : e+ 2∑

0≤i≤e−1

ν(i) ≤ y ≤ e+ 2∑0≤i≤e

ν(i),

define qr0(z) = 0 and ΘYr0(z) = 0 for all oracles Y . By recursion theorem,

h(z) ∈ C†[t] for all t ≥ s+ 1, and h(z) is the desired location marker.

54


(⋆⋆) Define the coding indicator and the coding marker. First choose a coding

indicator c ∈ ⟨3e+2,N⟩ which is larger than all numbers used so far, and

then pick the least unused number x in

⟨3e+ 1, y⟩ : e+ 2∑

0≤i≤e−1

ν(i) ≤ y ≤ e+ 2∑0≤i≤e

ν(i).

Define qr0(x) = c and ΘYr0(x) ↓⇔ c /∈ Y for all oracles Y . By recursion

theorem, h(x) ∈ C†[t] iff c /∈ Ct for all t ≥ s + 1. h(x) is the desired

coding marker with the coding indicator c.

(⋆ ⋆ ⋆) Define ∆C†g(e)(e). If χ(e, βe,s+1) = 1, define

∆C†g(e)(e)[s+ 1] = C†[s+ 1](h(x)) = 1− Cs+1(c) = 1.

Otherwise, χ(e, βe,s+1) = 0. First enumerate the coding indicator c into

Cs+1, so C†[s+ 1](h(x)) = 0, and then define

∆C†g(e)(e)[s+ 1] = C†[s+ 1](h(x)) = 1− Cs+1(c) = 0.

(2.3.2) ie,s+1 = ie,s and je,s+1 < je,s, the approximation of ∅††(e) changes at same

level. Let c be the uncanceled coding indicator at level ie,s, and h(x) be the

corresponding coding marker.

If χ(e, βe,s+1) = 1, then χ(e, βe,s) = 0. As

∆C†g(e)(e)[s] = C†[s](h(x)) = 1− Cs(c) = 0,

c ∈ Cs. Now remove c out of C, then

∆C†g(e)(e)[s+ 1] = C†[s+ 1](h(x)) = 1− Cs+1(c) = 1.

If χ(e, βe,s+1) = 0, then χ(e, βe,s) = 1, and

∆C†g(e)(e)[s] = C†[s](h(x)) = 1− Cs(c) = 1,

So c /∈ Cs. Now enumerate c into C. Then

∆C†g(e)(e)[s+ 1] = C†[s+ 1](h(x)) = 1− Cs+1(c) = 0.

55


Check whether there is a requirement Qj(e < j ≤ s + 1) which requires

attention at stage s + 1. If there is such a Qj, find the highest priority one,

act correspondingly. Otherwise, go to step (3).

(3) Defining ΓC⊕WC. Now we obtain Cs+1. Calculate C-restraints

r(i, s+ 1) = maxψC(i)[t] : t ≤ s+ 1 ∧ΨC(i)[t] ↓

on each Ri with i ≤ s+ 1.

– If there is a r(i, s+ 1) such that r(i, s+ 1) > r(i, s), then

i ∈ WC [s+ 1]−WC [s]

and ΨC(i)[s + 1] is defined with a new computation. If ΓC⊕WC(i)[s] ↓,

then according to a basic Pi-strategy, ΓC⊕WC(i) is undefined currently.

When we define ΓC⊕WC(i) at stage t ≥ s+ 1, its use γC⊕WC

(i)[t] will be

of the form ⟨3e, x⟩ with x larger than all numbers used so far, especially,

γC⊕WC(i)[t] > r(i, s+ 1).

Find the least y with ΓC⊕WC(y) ↑ currently.

If ΓC⊕WC(y) is never defined before,

– define ΓC⊕WC(y)[s+1] = ∅′[s+1](y) with use ⟨3e, x+1⟩, where x is the

least number larger than all numbers used so far.

Otherwise, let w < s+ 1 be the largest stage at which ΓC⊕WC(y)[w] ↓.

– If y ∈ ∅′[w + 1]−∅′[w], or

∃t ∈ [w, s] such that y ∈ WC [t+ 1]−WC [t] with r(y, t+ 1) > r(y, t), or

γC⊕WC(y)[w] ≤ γC⊕WC

(y − 1)[s],

then define ΓC⊕WC(y)[s+ 1] = ∅′[s+ 1](y) with use ⟨3e, x+ 1⟩, where x

is the least number larger than all numbers used so far.

56


– Otherwise, γC⊕WC(y)[w] > γC⊕WC

(y−1)[s], just define ΓC⊕WC(y)[s+1] =

∅′[s+ 1](y) with use γC⊕WC(y)[w].

This ends the construction of C at stage s+ 1.

3.1.6 Verification

Here, we verify the r0-th construction which is provided by the recursion theorem.

Recall that we have already defined computable function g(e), which is gr0(e), in

a basic Qe-strategy with g(e) ≥ δC†(e) = maxUe,r0 , where Ue,r0 is the set of injury

markers, location markers and coding markers of Qe defined in the r0-th construc-

tion.


(1) Q is injured at most finitely often;


(3) Q injures all lower priority strategies finitely often.

Proof: We will prove the lemma by induction on e that all Re, Pe, and Qe are

true.

For e = 0. R0 has the highest priority, it is not injured. (1) holds. If 0 ∈WC [s]−WC [s−1] at some first stage s, then r(0, s) = ψC(0)[s] and the computation

ΨC(0)[s] is never destroyed later [indeed, after stage s, all Pj and Qj with j ≥ 0 can

only deal with witnesses larger than r(0, s)]. Then 0 ∈ WC [t] for all t ≥ s. So R0

is satisfied, (2) holds. R0 acts at most once by setting C-restraints, thus initializes

lower priority strategies at most once. (3) holds.

For P0. P0 is injured by R0 at most once when 0 ∈ WC [s] − WC [s − 1] at

some first stage s, (1) holds. In this case, ΓC⊕WC(0) ↑, we will define it with a

big use later. Let w0 be the least stage such that P0 is not injured after stage w0.

After stage w0, P0 acts at most once if 0 ∈ ∅′[t] − ∅′[t − 1] at some stage t with

ΓC⊕WC(0)[t− 1] ↓. In this case, γC⊕WC

(0)[t− 1] is enumerated into Ct, and we will

define ΓC⊕WC(0) = 1 with a large use later. So γC⊕WC

(0) increases at most twice

57


after its first definition, there is a stage w1 ≥ w0 such that ΓC⊕WC(0)[w1] is defined

and is never undefined later. (2) holds. P0 never acts after stage w1. (3) holds.

For Q0. Recall that we use s0 to denote the first stage at which there is an

ordinal α < ω2 such that χ(0, α)[s0] ↓, and let β0,s0 = ω · is0 + js0 be the least

such ordinals. We will appoint a coding indicator c at level is0 , and then define

corresponding coding marker h(z). Define

∆C†g(0)(0)[s0] = C†[s0](h(z)) = 1− Cs0(c) = χ(0, β0,s0)

for first time.

When 0 ∈ WC [s]−WC [s− 1] at some first stage s ≥ s0 with r(0, s) bigger than

coding indicators of Q0, Q0 is injured. We will enumerate an injury marker into

C† to undefine all old definitions of ∆C†g(0)(0). Note P0 may act at most once by

enumerating γC⊕WC(0) into C, but this enumeration has no essential influences to

the definition of ∆C†g(0)(0) by the choice of Γ-uses and coding indicators. So Q0 is

injured at most once by R0. (1) holds.

Let t0 ≥ s0 be the least stage such that Q0 is not initialized after stage t0, and

let β0,t = ω · i0,t + j0,t be the least ordinal such that χ(0, β0,t)[t] ↓ at stage t ≥ s0.

After stage t0, if β0,s < β0,s−1 via i0,s < i0,s−1, then all the previous definitions of

∆C†g(0)(0) are injured at stage s, because we enumerate a location marker at level

i0,s into C†. Now assume that after stage t1 ≥ t0, i0,t never decreases, i.e., i0,t = i0,t1

for all t ≥ t1 (such a t1 exists since limsβ0,s exists).

Let c′ be the coding indicator at level i0,t1 and h(z′) be the corresponding coding

marker. We define h(z′) such that h(z′) ∈ C†[s] ⇔ c′ /∈ Cs for all stage s ≥ t1. At

stage t1, we maintain

∆C†g(0)(0)[t1] = C†[t1](h(z′)) = 1− Ct1(c

′) = χ(0, β0,t1).

When j0,t < j0,t−1 at some stage t > t1, we will enumerate c′ into Ct if χ(0, β0,t−1) =

1, and extract c′ out of Ct−1 if χ(0, β0,t−1) = 0, keeping

∆C†g(0)(0)[t] = C†[t](h(z′)) = 1− Ct(c′) = χ(0, β0,t).

Let t2 ≥ t1 be the least stage such that j0,t = j0,t2 for all t ≥ t2 (again, such a

t2 exists since limsβ0,s = β0; in this case, β0 = ω · i0,t1 + j0,t2). Then for all t ≥ t2,

58


β0,t = β0,t2 and ∅††(0) = χ(0, β0,t). So Q0 never acts after stage t2. Hence,

∆C†g(0)(0) = C†[t2](h(z′)) = 1− Ct2(c

′) = χ(0, β0,t2) = ∅††(0).

We have showed that Q0 is satisfied after finitely many actions, and thus injures

lower priority strategies finitely often. (2) and (3) hold.

Recall that sl is the first stage at which there is an ordinal α < ω2 such that

χ(l, α)[sl] ↓. Let βl,s = ω · il,s + jl,s be the least ordinal such that χ(l, βl,s)[s] ↓ at

stage s ≥ sl.

Assume that the lemma is true for Ri, Pi and Qi with i < e. Let ve > maxsi :i ≤ e be the least stage such that for i < e,

• limsr(i, s) = r(i, ve),

• ΓC⊕WC(i)[ve] is defined and is never undefined after stage ve,

• ∆C†g(i)(i)[ve] is defined with value χ(i, βi,ve) and is never changed later.

Then noRi with i < e has C-restraints r(i, s) increase after stage ve, limsγC⊕WC

(i)[s] =

γC⊕WC(i)[ve] for all i < e, and lim

sβi,s = βi,ve for all i < e.

For Re. As no Γ-uses or coding indicators of positive requirements are enumer-

ated into or removed out of C after stage ve, Re is not injured after stage ve. (1)

holds. After stage ve, there are two cases.

(i) r(e, s) never increases after stage ve. Then limsr(e, s) = r(e, ve), Re is satisfied

and never acts after stage ve.

(ii) There is a least stage t > ve at which e ∈ WC [t] − WC [t − 1] with new

computation. Then r(e, t) > r(e, s′) for all s′ ≤ t−1. During the construction,

no positive requirements with lower priority will enumerate numbers ≤ r(e, t)

into C or extract numbers ≤ r(e, t) from C after stage t, so ΨC(0)[t] is never

injured after stage t. Hence, e ∈ WC [s′] for all s′ ≥ t. Re acts at most once

after stage ve and is satisfied.

59


Hence, (2) and (3) hold.

Similar to P0 and Q0 below R0. After stage ve, for Pe and Qe below Re, Pe is

injured at most once, and acts at most once by enumerating γC⊕WC(e) into C when

e goes into ∅′; Qe is injured at most once, and will be satisfied after finitely many

actions. So the lemma holds for Pe and Qe.

Lemma 3.2 For all x, |s : Cs(x) = Cs+1(x)| ≤ x+ 1. Thus C is ω-c.e.

Proof: We can effectively check whether x is a Γ-use or a coding indicator or

not. If no, Cs(x) = 0 for all stages s.

Assume that x is a Γ(i)-use for some i. It enumerates into C at most once when

i goes into ∅′, so |s : Cs(x) = Cs+1(x)| ≤ 1.

Assume that x is a coding indicator of some Qe appointed at stage s. Consider

the least ordinal βe,s = ω ·ie,s+je,s such that χ(e, βe,s)[s] ↓. Then x > je,s, and x may

be enumerated into Cs. At stage t > s, Ct(x) = Ct−1(x) only if je,t < je,t−1 ≤ je,s.

So |s : Cs(x) = Cs+1(x)| ≤ 1 + je,s ≤ x.

By Lemma 3.1, ∅†† ≤wtt C†, ∅′ ≤T C ⊕WC , and WC ≤T ∅′. By Lemma 3.2,

C is ω-c.e. So C is a bounded-high set with C ⊕WC ≡T ∅′.

This completes the proof of Theorem 3.1.(1).

3.2 Pseudo-jump inversion via bounded-low c.e.

sets

To prove Theorem 3.1.(2), for the c.e. operator W , we need to build a bounded-low

c.e. set C (i.e., C† is ω-c.e.) such that C ⊕WC ≡T ∅′. As C will be c.e., C† has

the following natural computable approximations:

fC†(x, s) :=

1, if ∃n ≤ x[φn(x)[s] ↓ ∧ Φ

Cφn(x)x (x)[s] ↓];

0, otherwise.

We only need to add negative requirements of the form

Ne : |s : fC†(e, s) = fC†(e, s+ 1)| ≤ 2(e+ 1)2 + 1

60


to the standard pseudo-jump inversion theorem. Then C will be bounded-low.

We will build a functional Γ such that ΓC⊕WC= ∅′. If e enumerates into ∅′,

then rectify ΓC⊕WC(e) = 1 by enumerating γC⊕WC

(e) into C. To make WC ≤T ∅′

(i.e., WC is ∆02), as usual, we will preserve e ∈ WC by setting a C-restraint ψC(e)

if ΨC(e) ↓.When defining ΓC⊕WC

, the Γ-uses satisfy the following rules.

(i) For each e, and stage s, if ΓC⊕WC(e) is newly defined at stage s, its use

γC⊕WC(e)[s] is defined to be larger than all numbers used before, especially,

γC⊕WC(e)[s] > e.

(ii) For each e, and stage s, if ΓC⊕WC(e)[s] ↓, then ΓC⊕WC

(e)[t] is undefined at

some stage t > s only if

Ct γC⊕WC(e)[s] = Cs γC⊕WC

(e)[s] ∨ WC [t] γC⊕WC(e)[s] = WC [s] γC⊕WC

(e)[s] .

(iii) For e < e′ and stage s, if ΓC⊕WC(e)[s] and ΓC⊕WC

(e′)[s] are both defined, then

γC⊕WC

(e)[s] < γC⊕WC

(e′)[s].

(iv) For each e, there is a stage se such that for all s ≥ se, ΓC⊕WC

(e)[s] ↓= ∅′[s](e)

with use γC⊕WC(e)[s] = γC⊕WC

(e)[se].

This ensures that ∅′ = ΓC⊕WC, and thus ∅′ ≤T C ⊕WC.

As in the standard pseudo-jump inversion theorem, the definition of ΓC⊕WC(e)

is compatible with preserving WC(e).

Now consider the bounded-low requirement Ne. In order to preserve C†(e), at

each stage s, we need to preserve a C-restraint

rN (e, s) = maxφn(e)[s] : n ≤ e ∧ φn(e)[s] ↓.

If ΓC⊕WC(e)[s] ↓ with γC⊕WC

(e)[s] ≤ rN (e, s), Ne will act to enumerate γC⊕WC(e)[s]

into C, thus making ΓC⊕WC(e) undefined; in this case, ΓC⊕WC

(e) will be redefined

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with a new use > rN (e, s). Since rN (e, s) can move up at most e+ 1 times, Ne will

enumerate at most e+ 1 many such uses.

Now computations of the form ΦCφn(e)e (e)[s] ↓ (of course, φn(e)[s] ↓) for some n ≤

e are injured at stage s+1 only if there is some i ≤ e such that γC⊕WC(i)[s] ≤ φn(e)

is enumerated into Cs+1. Now ΓC⊕WC(i) ↑ currently, it will be defined with new use

> φn(e). So the enumeration of γC⊕WC(i) later will not injure a new computation

of the form ΦCφn(e)e (e), and the enumeration of γC⊕WC

(i)-uses can injure ΦCφn(e)e (e)

at most once for fixed i and n. Thus, approximations for C†(e) are injured at most

(e+ 1)2 many times. Note that each such an injury entails at most two changes of

fC†(e, s). So

|s : fC†(e, s) = fC†(e, s+ 1)| ≤ 2(e+ 1)2 + 1.

Hence, C† is ω-c.e., i.e., C is bounded-low.

This completes the description of basic strategies and possible interactions be-

tween them. We omit the remaining construction and verification, which are finite

injury arguments.

This completes the proof of Theorem 3.1.(2).

62

Part II

Degrees of orders on torsion-freeabelian groups

63

Chapter 4

Group-order-computable degreesand PA degrees

In Proposition 1.5, we have proved that PA degrees and degrees which are not

weakly group-order-computable coincide. Then all PA degrees are not group-order-

computable. In this chapter, we will show that the converse is also true, that is,

degrees which are not PA are group-order-computable.

Theorem 4.1 There is a computable torsion-free abelian group G with infinite rank

such that for any set A, G admits exactly two A-computable orders iff A is not PA-

complete.

For G above, as any computable set is not PA-complete, G admits exactly two

computable orders. Then for a set A which is not PA-complete, every A-computable

order on G is computable; for a set A which is PA-complete, A computes a noncom-

putable order on G .

Let a be a degree which is not group-order-computable. Then for the group G

in Theorem 4.1, a computes a noncomputable order on G . Thus, a is a PA degree.

Now we obtain the following characterizations of PA degrees.

Proposition 4.1 Let a be a Turing degree. The following are equivalent.

(1) a is a PA degree.

(2) a is not weakly group-order-computable.

(3) a is not group-order-computable.

64

Chapter 4. Group-order-computable degrees and PA degrees

Note that 0′ is the only c.e. degree which is PA. An immediate corollary of

Theorem 4.1 is:

Corollary 4.1 There is a computable torsion-free abelian group G with infinite rank

such that

(1) G has exactly two computable orders;

(2) for any c.e. degree a < 0′, every a-computable order on G is computable.

The following sections are devoted to prove Theorem 4.1.

4.1 Requirements

To prove Theorem 4.1, we will build a computable group G and a computable order

≤G on G to meet the following requirements.

P : G ∼= Q∞.

Q : ≤G and ≤∗G are exactly two computable orders on G .

R : For any set A,

(1) if ≤A is an A-computable order on G , then either ≤A=≤G , or ≤A=≤∗G ,

or A is PA-complete;

(2) if A is PA-complete, then there is an A-computable order on G which is

not computable.

Here, Q∞ is the direct sum of infinitely many copies of Q. ≤∗G denotes the reversal

order of ≤G , that is, for x, y ∈ G , x ≤∗G y ⇔ y ≤G x.

If three requirements above are satisfied, then G is a computable group with

exactly two computable orders. Moreover, for any set A, if A is not PA-complete,

then every A-computable order on G is computable, so there are exactly two A-

computable orders on G ; if A is PA-complete, then there are A-computable orders

which are not computable, so there are more than two A-computable orders on G .

65


4.2 Background

We now review a few concepts on binary strings which are needed for constructing

G . Let 2<ω be the set of binary strings. For each binary string σ, the symbol |σ|denotes the length of σ. For the empty string λ, |λ| = 0. For a nonempty string

σ, |σ| ≥ 1; in this case, write σ := σ(0) · · ·σ(|σ| − 1) and for 1 ≤ i ≤ |σ|, letσ i:= σ(0) · · · σ(i− 1). Set σ 0:= λ.

2<ω has a natural lexicographical order ≤lex defined as follows: 0 <lex 1, and

for σ, τ ∈ 2<ω, σ ≤lex τ if either there is an i < min(|σ|, |τ |) with σ i= τ i andσ(i) <lex τ(i); or σ ⊆ τ . Another well-known order, the length-lexicographical order

≤len−lex, on 2<ω is defined like this: for σ, τ ∈ 2<ω,

σ ≤len−lex τ ⇔ |σ| < |τ | ∨ (|σ| = |τ | ∧ σ ≤lex τ).

Figure 4.1: 2<ω ∪ 2<ω@

Let 2<ω@ be the set of symbols σ@ with σ a binary string. Define a lexicograph-

ical order ≤lex on 2<ω ∪ 2<ω@ based on 0 <lex @ <lex 1. In particular, for each

σ ∈ 2<ω, σ0 <lex σ@ <lex σ1. We can view 2<ω ∪2<ω@ as a ternary tree, where each

node σ ∈ 2<ω has three immediate extensions, namely σ0 <lex σ@ <lex σ1, while

each σ@ with σ ∈ 2<ω has no extensions. The ternary tree is roughly pictured in

Figure 4.1.

Now define a length-lexicographical order, say ≤len−lex, on 2<ω@ based on the

lexicographical order on 2<ω@. That is, for σ@ and τ@ in 2<ω@,

σ@ ≤len−lex τ@ ⇔ |σ| < |τ | ∨ (|σ| = |τ | ∧ σ@ ≤lex τ@).

66


4.3 The P-strategy

We will build a computable group G stage by stage such that G is isomorphic to

Q∞ via a ∆02-isomorphism. During the construction, we maintain a finite set Bs at

each stage s which serves as an approximation of an infinite basis B of G . Then

G ∼= ⊕b∈B

Qb ∼= Q∞.

Now we start to define G . For each σ@ ∈ 2<ω@, we introduce a base-element

qσ@ for G . qσ@ will be enumerated into the constructed group at stage |σ|. qσ@ has

two totally different fates.

(1) If the R-strategy adds a dependence relation of the form qσ@ = rqτ@ for some

positive rational r and some qτ@ with τ@ <len−lex σ@ at some stage t > |σ|,then qσ@ is said to be non-active at stage t. In this case, qσ@ /∈ Bt′ for all

t′ ≥ t.

(2) If no dependence relations of the form qσ@ = rqτ@ are added during stages in

[|σ|, t], qσ@ is said to be active at stage t. In this case, qσ@ ∈ Bt.

Hence, either qσ@ is active forever in which case qσ@ ∈ B, or qσ@ becomes non-active

at some stage > |σ| in which case qσ@ /∈ B.

Let H be the computable additive group ⊕σ∈2<ω

Qqσ@. Elements of H are just

finite Q-linear sums of elements in qσ@ : σ ∈ 2<ω. The constructed group G is

indeed a quotient group of H .

Fix a computable enumeration Hs of H such that at most one element is enu-

merated into Hs at stage s, and for each nonzero x0qσ0@ + · · · + xmqσm@ in Hs, it

satisfies xm = 0, and for each i ≤ m, |σi| ≤ s. Assume that H0 = 0H .

Construction of G = (G,+, 0G ).

At stage 0.

G0 = 0G , q@, where 0G is the zero element of G and q@ is the active base-

element at stage 0. B0 = q@.

At the end of stage s, Gs is the set containing all elements of G enumerated by

stage s. Bs is the set of active base-elements at stage s. Each nonzero g ∈ Gs can

67


be expressed as a unique Q-linear sum of the form

r0qσ0@ + · · ·+ rnqσn@

of active base-elements at stage s with rn = 0 and σ0@ <lex σ1@ · · · <lex σn@ (this

restriction is clear in next paragraph about computable orders). By convention, the

unique Q-linear sum of 0G is the empty sum.

We require σ0@ <lex σ1@ · · · <lex σn@ in the expression g = r0qσ0@+ · · ·+rnqσn@because we build the computable order ≤G such that the sign of g under ≤G is just

the sign of the nonzero rm with m least if such a rm exists, and 0 otherwise. That

is, ≤G is indeed a kind of lexicographical order (the construction of ≤G is clear in

the following Q-strategy).

At stage s+ 1.

Define domain Gs+1. There are two cases depending on whether the R-strategy

adds dependence relations.

Case 1. If no dependence relations are added among active base-elements in Bs

by the R-strategy at stage s+1, then each base-element in Bs is still active at stage

s+1. For each σ with length s+1, add a new base-element qσ@ into Bs+1. That is,

Bs+1 = Bs ∪ qσ@ : |σ| = s+ 1.

If Hs+1 = Hs, let Gs+1 = Gs∪Bs+1. If there is an element, x0qσ0@+· · ·+xmqσm@

say, in Hs+1 − Hs. Note that xm = 0, and for each i ≤ m, |σi| ≤ s+ 1.

(1) If there are qσi@s such that qσi@ is non-active at stage s + 1, just update the

Q-linear sum by replacing such qσi@s by active base-elements at stage s + 1

with corresponding scaling.

(2) Otherwise, each qσi@ in the Q-linear sum is active at stage s+ 1.

In both cases, the Q-linear sum has a new expression, say a0qτ0@ + · · ·+ anqτn@, as

a Q-linear sum of elements in Bs+1. If all ai with i ≤ n are 0, then the Q-linear

sum is indeed the zero element of G , just let Gs+1 = Gs ∪Bs+1. Otherwise, find the

largest m ≤ n with am = 0, there are two cases:

68


(1) If a0qτ0@ + · · · + amqτm@ is not an expression of elements enumerated into G

before, then let Gs+1 = Gs ∪Bs+1 ∪ a0qτ0@ + · · ·+ amqτm@.

(2) Otherwise, let Gs+1 = Gs ∪Bs+1.

Case 2. If there are two active base-elements, say qσ@, qτ@ in Bs, such that the

pair (σ@, τ@) is enumerated into link at stage s+ 1, i.e., the relation link(σ@, τ@)

holds at stage s+ 1 [here, link is a c.e. equivalence relation on 2<ω@ which will be

defined in the basic R-strategy].

We will add dependence relations of the form qη@ = rqρ@ with r > 0, ρ@ <len−lex

η@, i.e., |ρ@| < |η@| or |ρ@| = |η@| with ρ@ <lex η@ (the method for adding

dependence relations will be provided in the following R-strategy). In this case, qη@

is non-active at stage s + 1. Once such a dependence relation is added, we update

the expression of elements in Gs. The added relations ensure that for any g, h ∈ Gs,

if the expression of g, h at stage s is different, then the new expression of g, h at

stage s+ 1 is still different.

Now add new active base-elements qσ@ with |σ| = s+1 at stage s+1. Let Bs+1

be the set of active base-elements at stage s+ 1 (the concert definition of Bs+1 will

be clear in the following R-strategy).

If there is a new x enumerating into Hs+1, first update x as a Q-linear sum of

active base-elements at stage s+ 1, and then add this Q-linear sum into Gs+1 as in

case 1.

This ends the construction of Gs+1.

Define the partial addition +s+1 on Gs+1. Note that each element of Gs+1 has

a unique Q-linear sum of active base-elements in Bs+1. For x, y, z ∈ Gs+1, if the

addition of the unique Q-linear sum of x and y is the same as that of z, then let

x+s+1 y = z.

If we already defined x +s y = z at stage s, then x +s+1 y = z. Indeed, after

substituting one dependence relation of the form qη@ = rqρ@ into the old expression

of x, y, z, we obtain the new expression of them as a unique Q-linear sum of current

active base-elements respectively. It is a direct checking that the new expression

69


of them also satisfies the condition that the addition of the new expression of x, y

equals that of z.

This ends the construction of +s+1.

4.4 The Q-strategy

Due to the one-to-one correspondence between orders and positive cones on com-

putable groups, when building an order on a group, it is enough to specify the set of

positive elements or alternatively to specify the sign of elements in the group. We

build ≤G by specifying the sign of elements in G under ≤G . That is, for each g ∈ G ,

the sign of g is

sign≤G(g) :=

+1, if g >G 0G

0, if g = 0G

−1, if g <G 0G

Then the positive cone of ≤G is just g ∈ G : sign≤G(g) = +1 ∪ 0G .

At each stage s, we approximate ≤G through a type of lexicographical order on

Gs in the following sense. The sign of 0G is always 0. Now assign the sign of nonzero

elements in Gs. First, each nonzero element g ∈ Gs has a unique Q-linear sum of

active base-elements of the form

r0qσ0@ + r1qσ1@ + · · ·+ rnqσn@

with rn = 0 and σ0@ <lex σ1@ · · · <lex σn@. Second, for each such a nonzero g,

find the least m with rm = 0 in the expression, and then define the sign of g to be

the usual sign of rm as rational numbers (i.e., for r ∈ Q, if r > 0, it has sign +1; if

r = 0, it has sign 0; if r < 0, it has sign -1). During the construction, we maintain

the sign of elements unchanged.

Construction of sign≤G.

At stage 0.

G0 = 0G , q@. Let the sign of 0G be 0 and the sign of q@ be +1.

At the end of stage s, we have already appointed the sign of elements in Gs.

Now define the sign of elements in Gs+1.

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At stage s+ 1.

Case 1. If no dependence relations are added via the R-strategy, then the set of

active base-elements at stage s+ 1 is Bs ∪ qσ@ : |σ| = s+ 1. The sign of elements

in Gs is unchanged at stage s+ 1.

For each new added active base-element qσ@ with |σ| = s+1, let sign≤G(qσ@) =

+1. If there is one more added element, x say, in Gs+1. Then x is of the form

a0qσ0@ + a1qσ1@ + · · · + amqσm@ with am = 0 and σ0@ <lex σ1@ · · · <lex σn@ as a

unique Q-linear sum of active base-elements at stage s + 1. Find the least i ≤ m

such that ai = 0, and then let sign≤G(x) be +1 if ai is positive, and −1 otherwise.

Case 2. If there are two active base-elements, say qσ@, qτ@ in Bs, such that the

pair (σ@, τ@) is enumerated into link at stage s+ 1, i.e., the relation link(σ@, τ@)

holds at stage s+ 1, we will add dependence relations to satisfy the R-strategy. To

make sign≤Gcomputable, we will ensure that such dependence relations added at

stage s+ 1 will not change the sign of elements in Gs.

Without loss of generality, let σ@ <lex τ@. In order to preserve signs of elements

in Gs, the R-strategy first finds all qη@ ∈ Bs such that

(1) σ@ ≤lex η@ ≤lex τ@;

(2) there is an element g ∈ Gs and a nonzero coefficient r such that rqη@ appears

in the unique expression of g as a Q-linear sum of elements in Bs.

Now list all such η@s under the lexicographical order as

σ@ = ξ0@ <lex ξ1@ · · · <lex ξn−1@ <lex ξn@ = τ@.

By the definition of link, link(σ@, τ@) holds implies for all i, j ≤ n, link(ξi@, ξj@)

holds.

In the following R-strategy, we will add dependence relations of the form

qη@ = rqρ@

with proper r > 0 and ρ@ <len−lex η@, i.e., |ρ| < |η| or (|ρ| = |η| ∧ ρ@ <lex η@).

We start with qξ0@ and qξ1@. Now each g ∈ Gs has an expression of the form

g = · · ·+ rξ0qξ0@ + rξ1qξ1@ + · · · .

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Let

δ0@ :=

ξ0@, if ξ0@ <len−lex ξ1@ξ1@, if ξ1@ <len−lex ξ0@

and γ0@ ∈ ξ0@, ξ1@ − δ0@. We first add relations of the form

qγ0@ = c0qδ0@

with proper c0 > 0 such that the sign of elements in Gs does not change after

substituting this relation.

When ξ0@ <len−lex ξ1@, we add qξ1@ = c0qξ0@. Then g has a new expression

· · ·+ (rξ0 + rξ1c0)qξ0@ + 0qξ1@ + · · · .

(1) If rξ0 = 0, then the sign of rξ1 equals the sign of rξ1c0.

(2) If rξ0 = 0, then we will choose c0 such that the sign of rξ0 equals the sign of

rξ0 + rξ1c0.

Then the sign of g computed according to the new expression is the same as before.

When ξ1@ <len−lex ξ0@, we add qξ0@ = c0qξ1@. Now g has a new expression

· · ·+ 0qξ0@ + (rξ0c0 + rξ1)qξ1@ + · · · .

(1) If rξ0 = 0, then the expression of g does not change.

(2) If rξ0 = 0, then we will choose c0 such that the sign of rξ0 equals the sign of

rξ0c0 + rξ1 .

So the sign of g does not change after substituting the relation.

Now each element in Gs has a unique expression as a Q-linear sum of elements

in Bs−qγ0@ and there are no active base-elements qζ@ with δ0@ <lex ζ@ <lex ξ2@

in the expression.

We then deal with qδ0@ and qξ2@ in a similar manner as dealing with qξ0@ and

qξ1@. Continue this process until all qξi@ with i ≤ n are considered. Once adding a

dependence relation, we preserve the sign of elements in Gs. So the whole process

does not change the sign of elements in Gs.

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In general, for 1 ≤ i ≤ n − 1, define by induction that δi@ is the length-

lexicographically smaller element in δi−1@, ξi+1@ and that γi@ is the other element

in δi−1@, ξ[email protected] the set of active base-elements is

Bs+1 = (Bs − qγi@ : 0 ≤ i ≤ n− 1) ∪ qσ@ : |σ| = s+ 1.

The sign of each new active base-element qσ@ is +1, and the sign of the new added

element (if such an element exists) is determined as in case 1.

4.5 The R-strategy

The R-requirement says that for any set A,

(1) if the constructed group G has an A-computable order, say ≤A, with ≤A =≤G

and ≤A =≤∗G , then A is PA-complete;

(2) if A is PA-complete, then G has an A-computable order which is neither ≤G

nor ≤∗G .

4.5.1 Background on PA-complete sets

We will use an equivalent definition for a set being PA-complete that A is PA-

complete iff it computes a 0, 1-valued diagonally nonrecursive function, i.e., a

0, 1-valued function f such that for all x, if φx(x) ↓, then f(x) = φx(x).

For a binary tree T , an infinite binary string f is an infinite path on T if for all

n, its prefix f(0)f(1) · · · f(n), denoted by f n+1, is in T ; set f 0:= λ, the empty

string. We view f as a 0, 1-valued function (i.e., the characteristic function of

n ∈ N : f(n) = 1). We use [T ] to denote the set of infinite paths of T .

One can build a co-c.e. binary tree T (i.e., the complement of T is c.e.) such

that [T ] is just the set of 0, 1-valued diagonally nonrecursive functions. Then for

a set A, A is PA-complete iff it computes an infinite path of T . Now based on this

T , to satisfy R, we only need to meet the following R′:

R′: For any set A, if ≤A is an A-computable order on G which is neither ≤G nor

≤∗G , then A computes an infinite path on T . If f ∈ [T ], then f computes an

order on G which is not computable.

73


We now construct the co-c.e. tree T by stages.

The construction of T .

At stage 0. T0 = 2<ω, the full binary tree. F0 = ∅, the complement of T0.

At the end of stage s, we already obtained approximations Ts and Fs such that

Ts ∪ Fs = 2<ω.

At stage s+1. For each σ ∈ Ts of length ≤ s + 1, if there is a x < |σ| suchthat φx,s+1(x) is defined with value σ(x), then enumerate σ into Fs+1. Let Ts+1 =

2<ω − Fs+1.

This ends the construction of T at stage s+ 1.

Clearly, T = ∩s∈N

Ts is a co-c.e. set. For any binary string σ, σ ∈ T iff it satisfies

∀x < |σ| [ φx(x) ↓→ φx(x) = σ(x) ], then any initial segment of σ is also in T . So

T is a tree. Furthermore, f ∈ [T ] iff f is a 0, 1-valued diagonally nonrecursive

function.

By using infinite paths on T , we will define an equivalence relation on 2<ω@

(recall that 2<ω@ = σ@ : σ ∈ 2<ω). The lexicographical order ≤lex on 2<ω ∪2<ω@

is already defined in section 4.2, where for any binary string σ, σ0 <lex σ@ <lex σ1.

Now for f ∈ [2<ω] and σ@ ∈ 2<ω@,

• say that f is lexicographically less than σ@, denoted by f <lex σ@, if there

is an i < |σ| + 1 such that f i= σ@ i and f(i) <lex σ@(i); and σ@ is

lexicographically less than f , denoted by σ@ <lex f , otherwise.

Hence, if f <lex σ@, then for all n, f n<lex σ@; otherwise, there is a m such that

for all n ≥ m, σ@ <lex f n.

Definition 4.1 Let T be the co-c.e. tree, where [T ] is the set of 0, 1-valued diag-

onally nonrecursive functions. For σ@, τ@ ∈ 2<ω@, link(σ@, τ@) holds if there is

no f ∈ [T ] such that f is lexicographically between σ@ and τ@.

We use link to denote the set of pairs (σ@, τ@) such that link(σ@, τ@) holds.

One can directly check that link is an equivalence relation on 2<ω@.

Proposition 4.2 link is a c.e. equivalence relation.

74


Proof: Fix a computable approximation Ts of T (as in the construction of T

above) such that Ts+1 ⊆ Ts and T = ∩s∈N

Ts.

For σ@, τ@ ∈ 2<ω@, assume that σ@ <lex τ@. link(σ@, τ@) holds iff

η ∈ 2<ω : σ@ <lex η <lex τ@ ∩ T is finite iff

∃ n ∃ s ∀ ρ ∈ 2<ω [ (|ρ| = n ∧ σ@ <lex ρ <lex τ@) → ρ /∈ Ts ].

So link is Σ01.

4.5.2 The R′-strategy

The R′-requirement says that:

(1) for any set A, if ≤A is an A-computable order on G which is neither ≤G nor

≤∗G , then A computes an infinite path on T ;

(2) if f ∈ [T ], then f computes an order on G which is not computable.

In order to satisfy (1) of the R′-requirement, we will initiate an A-effective

process for searching nodes in 2<ω@ which approximates an infinite path on T

lexicographically form below and above. Our method is to add dependence relations

of the form qη@ = rqρ@, where r > 0, link(η@, ρ@) holds, and ρ@ <len−lex η@.

In this section, we will develop the method for adding dependence relations. We

will show that this method works (i.e., the R′-requirement is satisfied) in sections

4.6.3 and 4.6.4.

Since the length-lexicographical order on 2<ω@ is a well ordering, for each τ@ ∈2<ω@, there are finitely many ζ@ ∈ 2<ω@ such that ζ@ <len−lex τ@. So there is

a least ζ@ ≤len−lex τ@ such that ζ@ and τ@ are linked, where the least is taking

under ≤len−lex. Then qτ@ has an expression of the form r′qζ@ for some r′ > 0, and

then no new expressions are assigned to qτ@. Since an element, x say, is enumerated

into Gs for first time is always of the form

x0qσ0@ + · · ·+ xmqσm@

of a Q-linear sum of active base-elements at stage s, there is a large enough stage

after which no more expressions are assigned to all qσi@ with 0 ≤ i ≤ m. Then x

will arrive at a final expression as a Q-linear sum of active base-elements.

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When adding qη@ = rqρ@ at stage s + 1, we need to ensure the following two

conditions:

(1) the assignment of Q-linear sums to elements in Gs remains one-to-one (i.e.,

the dependence relation is compatible with the P-strategy of constructing G );

(2) the sign of elements in Gs does not change (i.e., the dependence relation is

compatible with the Q-strategy of constructing ≤G ).

Adding dependence relations.

Since link is a c.e. equivalence relation on 2<ω@, just fix a computable enumer-

ation links of link such that at each stage s there is at most one pair of the form

(σ@, τ@) with |σ|, |τ | < s enumerating into links.

At stage 0. link0 = ∅. q@ is the only active base-element at stage 0.

Assume that we have already obtained Bs, the set of active base-elements at

stage s. Each element in Bs is of the form qτ@ with |τ | ≤ s. Each g ∈ Gs has a

unique Q-linear sum of elements in Bs. By convention, the unique Q-linear sum for

0G is just the empty sum.

At stage s+1.

Case 1. If links = links+1, or (σ@, τ@) ∈ links+1 − links, where at least one

of qσ@ and qτ@ is non-active at stage s, then no dependence relations are added at

stage s+ 1. Just let Bs+1 = Bs ∪ qσ@ : |σ| = s+ 1.

Case 2. Suppose that there is a pair (σ@, τ@) of active base-elements at stage

s enumerating into links+1. We will add dependence relations between qσ@ and qτ@.

Without loss of generality, we assume that σ@ <lex τ@.

First, find all qη@ ∈ Bs such that

(1) σ@ ≤lex η@ ≤lex τ@;

(2) there is an element g ∈ Gs and a nonzero coefficient r such that rqη@ appears

in the unique expression of g as a Q-linear sum of elements in Bs.

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Second, list all such η@s under the lexicographical order as follows:

σ@ = ξ0@ <lex ξ1@ · · · <lex ξn−1@ <lex ξn@ = τ@.

Since link(σ@, τ@) holds, for all i, j ≤ n, link(ξi@, ξj@) holds.

We start to add dependence relations between qξ0@ and qξ1@. Now each element

of Gs has a unique Q-linear sum in which no active base-elements qζ@ with ξ0@ <lex

ζ@ <lex ξ1@ appears. That is, each g ∈ Gs has a unique expression of the form

g = · · ·+ rξ0qξ0@ + rξ1qξ1@ + · · · .Let δ0@ be the length-lexicographically less element between ξ0@ and ξ1@, and

γ0@ be the other element. In module D(0), we will add a relation of the form

qγ0@ = c0qδ0@ with proper c0 > 0 such that after substituting this relation, the new

expressions of elements in Gs as Q-linear sums of elements in Bs − qγ0@ remain

one-to-one and the signs of elements in Gs computed according to new expressions

remain unchanged.

Module D(0).

1. List all nonzero r ∈ Q where either rqδ0@ or rqγ0@ occurs in the unique expres-

sion of an element in Gs as a Q-linear combination of active base-elements at

stage s as follows:

r1, r2, · · · , rl.

2. Set r0 := 0. Choose a large enough k ∈ N+ such that for all i, i′, j, j′ ≤ l with

rj − rj′ = 0,

k >|ri − ri′||rj − rj′|

.

2.1. If δ0@ <lex γ0@, each element, say g ∈ Gs, has an expression of the form

g = · · ·+ rδ0qδ0@ + rγ0qγ0@ + · · · .

(1) Add a new dependence relation qγ0@ = 1kqδ0@. Declare qγ0@ is non-active

at stage s+ 1.

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(2) We now show that adding qγ0@ = 1kqδ0@ does not change the sign of

elements in Gs.

By substituting qγ0@ = 1kqδ0@, g has a new expression

g = · · ·+ (rδ0 +rγ0k)qδ0@ + 0qγ0@ + · · · .

When rδ0 = 0. Since k is positive, the sign of rγ0 equals the sign ofrγ0k.

When rδ0 = 0. We show that the sign of rδ0 equals the sign of rδ0 +rγ0k.

(i) rδ0 > 0. Since k > 0, if rγ0 ≥ 0, then rδ0 +rγ0k> 0. If rγ0 < 0, by the

choice of k, k >−rγ0rδ0

, that is, rδ0 +rγ0k> 0.

(ii) rδ0 < 0. Since k > 0, if rγ0 ≤ 0, then rδ0 +rγ0k< 0. If rγ0 > 0, since

k >rγ0−rδ0

, rδ0 +rγ0k< 0.

By definition, the current sign of nonzero g ∈ Gs is the sign of the first

nonzero rational coefficient in this new expression as a Q-linear sum of

current active base-elements. Hence, after adding the dependence relation

qγ0@ = 1kqδ0@, the current sign of g ∈ Gs is the same as before.

(3) We now show that adding qγ0@ = 1kqδ0@ ensures that the assignment of

Q-linear sums to elements in Gs remains one-to-one.

For x, x′ ∈ Gs, suppose that at stage s, x = · · ·+ aδ0qδ0@ + aγ0qγ0@ + · · · ,and x′ = · · ·+ a′δ0qδ0@ + a′γ0qγ0@ + · · · . If the new expression of x and x′

is the same, then the coefficient of active base-elements at stage s other

than qδ0@, qγ0@ is the same and

aδ0 +aγ0k

= a′δ0 +a′γ0k,

i.e., (aδ0 − a′δ0)k = (a′γ0 − aγ0).

(i) If aδ0 = a′δ0 , then a′γ0

= aγ0 . In this case, the old expression of x, x′

at stage s is the same.

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(ii) If aδ0 = a′δ0 , then k =a′γ0−aγ0aδ0−a

′δ0

. In this case, the old expression of x, x′

at stage s is different. However, by the choice of k, k >|a′γ0−aγ0 ||aδ0−a

′δ0

| . So

this case is impossible.

Hence, if x, x′ ∈ Gs have different expression as Q-linear sums of active

base-elements at stage s, then the new expression of them as Q-linear

sums of current active base-elements is still different.

2.2. If γ0@ <lex δ0@, each element, say g ∈ Gs, has an expression of the form

g = · · ·+ rγ0qγ0@ + rδ0qδ0@ + · · · .

(1) Add a new dependence relation qγ0@ = kqδ0@. Declare qγ0@ is non-active

at stage s+ 1.

(2) We first show that adding qγ0@ = kqδ0@ does not change the sign of

elements in Gs.

By substituting qγ0@ = kqδ0@, g has a new expression

g = · · ·+ 0qγ0@ + (rδ0 + rγ0k)qδ0@ + · · · .

If rγ0 = 0, then the expression of g does not change.

Suppose rγ0 = 0. There are two cases:

(i) rγ0 > 0. If rδ0 ≥ 0, then rδ0 + rγ0k > 0. If rδ0 < 0, since k >−rδ0rγ0

,

rδ0 + rγ0k > 0.

(ii) rγ0 < 0. If rδ0 ≤ 0, then rδ0 + rγ0k < 0. If rδ0 > 0, since k >rδ0−rγ0

,

rδ0 + rγ0k < 0.

The current sign of g is calculated as the sign of the first nonzero rational

coefficient in this new expression of g if such a nonzero coefficient exists.

Hence, the current sign of g ∈ Gs is the same as before.

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(3) We now show that adding qγ0@ = kqδ0@ ensures that the assignment of

Q-linear sums to elements in Gs remains one-to-one.

For x, x′ ∈ Gs with different expression x = · · ·+ aγ0qγ0@ + aδ0qδ0@ + · · ·and x′ = · · ·+ a′γ0qγ0@ + a′δ0qδ0@ + · · · . Suppose that the new expression

of x and x′ by substituting qγ0@ = kqδ0@ is the same.

Then the coefficient of active base-elements at stage s other than qγ0@, qδ0@

is the same and

aδ0 + aγ0k = a′δ0 + a′γ0k,

i.e., (aδ0 − a′δ0) = (a′γ0 − aγ0)k.

(i) If aδ0 = a′δ0 , then a′γ0

= aγ0 . In this case, the old expression of x, x′

at stage s is the same, contracting with the assumption on x, x′.

(ii) If aδ0 = a′δ0 , then k =aδ0−a

′δ0

a′γ0−aγ0. However, by the choice of k, k >

|aδ0−a′δ0

||a′γ0−aγ0 |

. So this case is impossible.

Hence, the new expression of x, x′ ∈ Gs as Q-linear sums of elements in

Bs − qγ0@ is different.

3. If n > 1, run module D(1). Otherwise, n = 1 and ξ1 = τ , stop adding

dependence relations.

This completes the description of module D(0).

At the end of module D(0), each element in Gs has a unique Q-linear sum of

elements in Bs − qγ0@. Now there is no current active base-elements of the form

qζ@ with δ0@ <lex ζ@ <lex ξ2@.

In general, when n ≥ i+1 ≥ 2, we will start a new module D(i). Assume that we

have already defined δi−1@ and γi−1@, and added dependence relations qγj@ = cjqδj@

for all j ≤ i − 1. Now each element in Gs has a unique Q-linear sum of elements

in Bs − qγj@ : j ≤ i − 1, and there are no current active base-elements qζ@ with

δi−1@ <lex ζ@ <lex ξi+1@.

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Let δi@ be the length-lexicographically smaller element in δi−1@, ξi+1@ and γi@be the other element. We will add a dependence relation of the form qγi@ = ciqδi@

in module D(i).

Module D(i) with 1 ≤ i ≤ n− 1.

1. List all nonzero rational numbers which are coefficients of qδi@ or qγi@ in the

unique expression of an element in Gs as a Q-linear sum of elements in Bs −qγj@ : j ≤ i− 1 as follows:

r1, r2, · · · , rl.

2. Set r0 := 0. Choose a large enough k ∈ N+ such that for all i, i′, j, j′ ≤ l with

rj − rj′ = 0,

k >|ri − ri′||rj − rj′|

.

If δi@ <lex γi@, add a new dependence relation qγi@ = 1kqδi@. If γi@ <lex δi@,

add a new dependence relation qγi@ = kqδi@. Declare qγi@ is non-active at

stage s+ 1.

By substituting the added dependence relation in the expression of g ∈ Gs

as a Q-linear sum of elements in Bs − qγj@ : j ≤ i − 1, we obtain a new

expression of g as a Q-linear sum of elements in Bs − qγj@ : j ≤ i. One can

show as in module D(0) that the sign of g determined by this new expression is

unchanged and that the new expression of elements in Gs remains one-to-one.

3. When n = i + 1, ξi+1 = τ , stop adding dependence relations. Otherwise,

n > i+ 1, continue to run module D(i+ 1).

This completes the description of module D(i).

Suppose that we have added dependence relations qγi@ = ciqδi@ in module D(i)

for all i ≤ n− 1. By a direct checking, one can show that qγi@ = aiqδn−1@ for some

ai > 0. Finally, the set of active base-elements at stage s+ 1 is

Bs+1 = (Bs − qγi@ : 0 ≤ i ≤ n− 1) ∪ qσ@ : |σ| = s+ 1.

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This ends adding dependence relations.

We will show in the following sections 4.6.3 and 4.6.4 that for any set A, G

admits exactly two A-computable orders iff A is not PA-complete.

4.6 Construction and verification

4.6.1 Construction

Fix an effective enumeration links of link (a c.e. equivalence relation defined on

2<ω@) such that at most one pair is enumerated into links at each stage s. Assume

that |σ| < s and |τ | < s when (σ@, τ@) ∈ links.

At stage 0. link0 = ∅. Let G0 = 0G , q@, where 0G is the zero of the constructed

group (its unique Q-linear sum is just the empty sum), q@ is the active base-element

at stage 0. Set sign≤G(0G ) = 0 and sign≤G

(q@) = +1.

At stage s+ 1.

Step 1. If a new pair of active base-elements at stage s is enumerated into

links+1, then add dependence relations according to the basic R-strategy, goto step

2. Otherwise, no dependence relations are added at stage s+ 1, goto step 3.

Step 2. Build Gs+1 and sign≤Gon Gs+1 respectively according to the basic

P-strategy and the basic Q-strategy with dependence relations added.

Step 3. Build Gs+1 and sign≤Gon Gs+1 respectively according to the basic

P-strategy and the basic Q-strategy with no dependence relations added.

This ends the stage s+ 1 of the construction.

In section 4.6.2, we will show that P and Q are both satisfied. In sections 4.6.3

and 4.6.4, we will show that R is satisfied.

4.6.2 G is a computable group possessing computable orders

Fix an effective enumeration Hs of H :=⊕

σ∈2<ω

Qqσ@ such that for all s, |Hs+1 −

Hs| ≤ 1. During the stage s of the construction, we enumerate qσ@ with |σ| = s

into Gs, and obtain active base-elements Bs. We may enumerate one more element,

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say x, into Gs; in this case, there is an element y in Hs such that x is a unique

Q-linear sum of elements in Bs obtained by replacing non-active base-elements in y

by active ones with corresponding scaling, and no elements enumerated into G so

far has an expression the same as x.

For each g ∈ Gs, it can be expressed as a unique Q-linear combination of active

base-elements at stage s. Now we show that the expressions of g only change finitely

many times after stage s. Then each element in G has a unique Q-linear combination

of active base-elements at end.

We only need to show that each original base-element, say qσ@, has finitely many

expressions (all such expressions are elements of H , they represent the same element

of G ). qσ@ is first enumerated into G as an active base-element at stage |σ|, it has anew expression later only if there is a τ@ <len−lex σ@ such that link(σ@, τ@) holds

(in this case, qσ@ will be non-active forever). Then qσ@ = aqτ@ for some a > 0.

Since ≤len−lex is a well ordering on 2<ω@, there is a length-lexicographically

least element, say ρ@, in 2<ω@ such that link(σ@, ρ@) holds. Then qσ@ has a last

expression of the form bqρ@ for some b > 0. In this case, qρ@ is active forever.

For each g ∈ Gs, we also define the sign sign≤G(g) such that if g has a new

expression later, then the sign computed according to this new expression is the

same as before. Now for each element, g say, in G , it may have several expressions

in terms of elements in H ; however, sign≤G(g) is just the sign computed according

to the first expression of g.

G is computable. For two original elements, say g, h, in H , one can effectively

check whether they represent the same element in G (i.e. g equals h in G ) as follows:

(1) find the stage t such that g − h is first enumerated into H ; (2) at stage t, we

will consider g − h and update it as a Q-linear sum, say x, of elements in Bt; (3)

g − h = 0G iff all coefficients in x are 0. Hence, this equality relation on H is

computable. G can be viewed as the quotient group of H modulo this equality

relation. Thus, G is computable.

≤G is computable. Now sign≤G(g) is a computable function on G . Moreover,

P := g ∈ G : sign≤G(g) = +1 ∪ 0G is a computable set which is closed under

addition, containing exactly one of g and −g when g = 0G . That is, P is a positive

cone on G . So P determines a computable order, namely ≤G , on G .

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G is ∆02-isomorphic to Q∞. Since there are infinitely many infinite paths on

T , the so defined link-relation on 2<ω@ has infinitely many equivalence classes, that

is, B := qσ@ : σ ∈ 2<ω, qσ@ is active is infinite. Moreover, B forms a Π01-basis for

G , and G is isomorphic to ⊕b∈B

Qb via a ∆02-isomorphism. Hence, G is ∆0

2-isomorphic

to Q∞.

4.6.3 A-computable orders on G which are not computable

Suppose that ≤A is an A-computable order on G such that ≤A =≤G and ≤A =≤∗G .

We will show that A computes an infinite path on the co-c.e. binary tree T which

is PA-complete. Then A is also PA-complete.

Note that G is just H modulo the so defined computable equality relation,

denoted by ∼. For any order on G with corresponding sign function, say φ, we can

indeed extend φ to H by setting φ(h) = φ(g) where g ∈ G such that g ∼ h. In

particular, we will talk about the sign of elements in H under ≤A or ≤G through

this way.

We start by comparing ≤A with ≤G on qσ@ : σ ∈ 2<ω, the set of original

base-elements. The sign of each qσ@ under ≤G is +1. For the sign determined by

≤A or simply called A-sign, either there are two base-elements, say qσ@ and qτ@,

having different A-sign; or all base-elements have the same A-sign.

Case 1. Suppose that there are qσ@ and qτ@ such that their A-sign are different.

Since each base-element is nonzero in G , its A-sign is not 0. Suppose that the A-sign

of qσ@ is -1, and that the A-sign of qτ@ is +1.

Claim. link(σ@, τ@) fails.

Proof. Suppose otherwise. We will add a dependence relation of the form qσ@ =

rqτ@ with r > 0, τ@ <len−lex σ@ or of the form qτ@ = r′qσ@ with r′ > 0, σ@ <len−lex

τ@. In both cases, qτ@ and qσ@ have the same A-sign.

This ends the proof.

Recall that σ@ and τ@ are not linked means that there is at least one infinite

path on T such that it is lexicographically between σ@ and τ@.

With the help of ≤A, we will initiate an A-effective process which approximates

lexicographically form below and above an infinite path of T . Without loss of

generality, assume that τ@ <lex σ@.

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Let ξ0@ = τ@, ζ0@ = σ@. By assumption, qξ0@ and qζ0@ have A-sign +1 and

-1 respectively. Let δ0 be the longest common initial segment of ξ0, ζ0, denoted

by ξ0 ∩ ζ0. Then any infinite binary string lexicographically between ξ0@ and ζ0@

extends δ0. Start module C(0) below, where one can determine a strictly extension

δ1 of δ0 such that δ1 is a prefix of the desired infinite path (we are searching for) on

T .

In general module C(i) with i ≥ 0, we define δi+1 ⊃ δi such that δi+1 can be

extended into an infinite path on T . C(i) proceeds as follows.

Module C(i) with i ≥ 0.

1. Pick the length-lexicographically least element η ∈ 2<ω with ξi@ <lex η@ <lex

ζi@, denoted by ηi.

2. If qηi@ has A-sign -1, then update ξi+1@ = ξi@, ζi+1@ = ηi@.

If qηi@ has A-sign +1, then update ξi+1@ = ηi@, ζi+1@ = ζi@.

In both cases, qξi+1@ and qζi+1@ have different A-sign. So there is an infinite

path on T which is lexicographically between ξi+1@ and ζi+1@.

3. Now define δi+1 ⊃ δi such that there is an infinite binary string on T extending

δi+1. There are three cases.

3.1. If ξi@ <lex δi@ <lex ζi@, then δi is the length-lexicographically least

element η in 2<ω such that ξi@ <lex η@ <lex ζi@, i.e., ηi = δi.

(i) If ζi+1 = ηi, then any infinite binary string lexicographically between

ξi+1@ and ζi+1@ extends δi0. Just set δi+1 = δi0.

(ii) If ξi+1 = ηi, then any infinite binary string lexicographically between

ξi+1@ and ζi+1@ extends δi1. Just set δi+1 = δi1.

3.2. If δi@ ≤lex ξi@, then δi1 ⊆ ηi. In this case, any infinite binary string

lexicographically between ξi+1@ and ζi+1@ extends δi1. So set δi+1 = δi1.

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3.3. If ζi@ ≤lex δi@, then δi0 ⊆ ηi. In this case, any infinite binary string

lexicographically between ξi+1@ and ζi+1@ extends δi0. So set δi+1 = δi0.

As link(ξi+1@, ζi+1@) fails, there is an infinite path, say f , on T which is

lexicographically between ξi+1@ and ζi+1@. Since any infinite binary string

lexicographically between ξi+1@ and ζi+1@ extends δi+1, f extends δi+1.

4. Start module C(i+ 1).

This completes module C(i).

We obtain a strictly increasing sequence δi : i ∈ N of binary strings such that

each δi can be extended into an infinite path on T . Let g = ∪i∈Nδi, i.e., g(n) = δn+1(n)

(note that |δn+1| ≥ n + 1). Then g is the desired infinite path on T which is

computable from ≤A, thus g is A-computable. Since g is PA-complete, A is also

PA-complete.

Case 2. Suppose that each original base-element has the same A-sign under

≤A. Suppose that this A-sign is +1, we will show that ≤A can compute an infinite

path of T [otherwise, we can deal with the reversal order of ≤A, and then show that

this reversal order can compute an infinite path of T ].

Now for each qσ@, its A-sign is the same as sign≤G(qσ@). Since ≤A =≤G , there is

an element, say g ∈ G, such that sign≤G(g) = +1, but the A-sign of g is -1. g has

a final Q-linear sum of active base-elements, denoted by g = a0qσ0@ + · · · + anqσn@

with ai = 0 for each i = 0, · · · , n and σ0@ <lex σ1@ · · · <lex σn@.

As sign≤G(g) = +1, a0 > 0. We may assume a0 = 1 [otherwise we deal with

1a0g]. Now write g as

g = [qσ0@ + (a1 − 1)qσ1@] + · · ·+ [qσm−1@ + (am − 1)qσm@] + · · ·+ qσn@.

For each 1 ≤ m ≤ n, sign≤G(qσm−1@+(am−1)qσm@) = +1, and sign≤G

(qσn@) = +1.

Note that qσn@ has A-sign +1. Since the A-sign of g is -1, there is some m ≤ n

such that qσm−1@ + (am − 1)qσm@ has A-sign -1. In this case, am − 1 < 0; otherwise,

am − 1 ≥ 0, then qσm−1@ + (am − 1)qσm@ has A-sign +1, a contradiction.

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Hence, there are σ@, τ@ with σ@ <lex τ@ and r > 0 such that qσ@ − rqτ@ has

sign +1 under ≤G and -1 under ≤A respectively.

Claim. For such σ@, τ@, link(σ@, τ@) fails.

Proof. Suppose otherwise. There are two cases:

If τ@ <len−lex σ@, then a dependence relation of the form qσ@ = aqτ@ with a > 0

will be added such that 0qσ@ + (a− r)qτ@ has sign +1 under ≤G . So a− r > 0.

In this case, since the A-sign of qτ@ is +1, the A-sign of 0qσ@ + (a− r)qτ@ is +1.

As qσ@−rqτ@ = 0qσ@+(a−r)qτ@ in G , if ≤A is an order on G , by our convention, the

A-sign of qσ@−rqτ@ and 0qσ@+(a−r)qτ@ is the same, contradicting the assumption

that qσ@ − rqτ@ has A-sign -1.

If σ@ <len−lex τ@, then we will add a dependence relation of the form qτ@ = bqσ@

with b > 0 such that sign≤G((1 − rb)qσ@ + 0qτ@) = +1. So 1 − rb > 0. Then

(1− rb)qσ@ + 0qτ@ has A-sign +1, a contradiction.

This ends the proof.

Similar to case 1, we will initiate an A-effective process which approximates an

infinite path on T lexicographically form below and above.

Find (σ@, τ@) with σ@ <lex τ@ and r > 0 such that qσ@−rqτ@ has sign +1 under

≤G and sign -1 under ≤A. Let ξ0@ = σ@, ζ0@ = τ@ and r0 = r. Note that any

infinite binary string lexicographically between ξ0@ and ζ0@ extends δ0 := ξ0 ∩ ζ0.First start module C(0), where we will determine the successor, say j, of δ0 such

that there is an infinite path on T which extends δ0j.

In general, for i ≥ 1, suppose that we have already obtained ξi@ <lex ζi@,

ri > 0, and δi in module C(i− 1), where sign≤G(qξi@ − riqζi@) = +1, the A-sign of

qξi@− riqζi@ is -1, and there is an infinite binary string on T which extends δi. Then

we will start module C(i) to search for longer approximation δi+1 of an infinite path

on T . We point out here that for i ≥ 1, δi may not be ξi ∩ ζi, but the point is that

any infinite binary string lexicographically between ξi@ and ζi@ extends δi.

Module C(i) with i ≥ 0.

1. Pick the length-lexicographically least element η ∈ 2<ω such that ξi@ <lex

η@ <lex ζi@, denoted by ηi.

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2. Search until a positive r′ is found such that qξi@− r′qηi@ and qηi@− rir′qζi@ have

the same sign +1 under ≤G ; and either qξi@ − r′qηi@ or qηi@ − rir′qζi@ has sign

-1 under ≤A.

We now show that such a r′ exists. Indeed, since link(ξi@, ζi@) fails, there is

an infinite binary string, say f , on T which is lexicographically between ξi@

and ζi@. There are three cases:

2.1. If ξi@ <lex f <lex ηi@ [so link(ξi@, ηi@) fails], and link(ηi@, ζi@) holds.

(i) ηi@ <len−lex ζi@. Then qζi@ = aqηi@ for some a > 0, and

qηi@ − rir′qζi@ = (1− ria

r′)qηi@ + 0qζi@.

One can choose a positive r′ such that 1− riar′> 0.

(ii) ζi@ <len−lex ηi@. Then qηi@ = bqζi@ for some b > 0, and

qηi@ − rir′qζi@ = 0qηi@ + (b− ri

r′)qζi@.

One can choose a positive r′ such that b− rir′> 0.

In both cases, qηi@ − rir′qζi@ has sign +1 under both ≤G and ≤A.

Since link(ξi@, ηi@) fails (no relation will be added between qξi@ and

qηi@), qξi@ − r′qηi@ has sign +1 under ≤G .

As the A-sign of qξi@ − riqηi@ is -1, the A-sign of qηi@ − rir′qζi@ is +1, and

qξi@ − riqηi@ = (qξi@ − r′qηi@) + r′(qηi@ − rir′qζi@),

the A-sign of qξi@ − r′qηi@ is -1.

2.2. If ηi@ <lex f <lex ζi@ [so link(ηi@, ζi@) fails], and link(ξi@, ηi@) holds.

(i) ξi@ <len−lex ηi@. Then qηi@ = aqξi@ for some a > 0, and

qξi@ − r′qηi@ = (1− r′a)qξi@ + 0qηi@.

One can choose a positive r′ such that 1− r′a > 0.

88


(ii) ηi@ <len−lex ξi@. Then qξi@ = bqηi@ for some b > 0, and

qξi@ − r′qηi@ = 0qξi@ + (b− r′)qηi@.

One can choose a positive r′ such that b− r′ > 0.

Similarly, one can show that qξi@ − r′qηi@ and qηi@ − rir′qζi@ have sign +1

under ≤G ; and that qξi@ − r′qηi@ and qηi@ − rir′qζi@ have A-sign +1 and -1

respectively.

2.3. If link(ξi@, ηi@) and link(ηi@, ζi@) both fail. Clearly, qξi@ − qηi@ and

qηi@ − riqζi@ have sign +1 under ≤G . Since the A-sign of qξi@ − riqζi@ is

-1, and

qξi@ − riqζi@ = (qξi@ − qηi@) + (qηi@ − riqζi@),

at least one of qξi@ − qηi@ and qηi@ − riqζi@ has A-sign -1. In this case,

just pick r′ = 1.

3. If the A-sign of qξi@− r′qηi@ is -1, update ξi+1@ = ξi@, ζi+1@ = ηi@, ri+1 = r′.

If the A-sign of qηi@− rir′qζi@ is -1, update ξi+1@ = ηi@, ζi+1@ = ζi@, ri+1 =

rir′.

In both cases, sign≤G(qξi+1@ − ri+1qζi+1@) = +1, but the A-sign of qξi+1@ −

ri+1qζi+1@ is -1. Hence, link(ξi+1@, ζi+1@) fails. There is an infinite path on

T which is lexicographically between ξi+1@ and ζi+1@.

4. Define δi+1 ⊃ δi such that there is an infinite path on T extending δi+1 as in

the step 3 of module C(i) in case 1.

5. Start module C(i+ 1).

This completes module C(i).

After running all modules C(i) with i ≥ 0, we arrive at an increasing sequence

δi : i ∈ N such that there is an infinite path on T extending each δi. Let g = ∪i∈Nδi.

Then g is an A-computable path on T , thus, A is PA-complete.

In sum, we have showed that every order on G is either computable or computable

form a PA-complete set. Hence, if A is not PA-complete, then every A-computable

on G is computable.

89


4.6.4 Orders on G which are computable from PA-completesets

Recall that the set of infinite paths of the co-c.e. tree T is just the set of 0, 1-valued diagonally nonrecursive functions. Let A be any PA-complete set. Then A

computes an infinite path on T . In order to prove that there is an A-computable

order on G which is not computable, we only need to show that any infinite path

on T computes a noncomputable order on G .

Let f ∈ [T ]. Now build a f -sign on G which determines a f -computable order,

≤f say, on G . For each original base-element qσ@,

(1) if σ@ <lex f , then set the f -sign of qσ@ to be +1;

(2) if f <lex σ@, then set the f -sign of qσ@ to be -1.

We then build the f -sign of ≤f by a similar algorithm as in building the sign of ≤G .

For instance, assume that x ∈ Gs has a unique Q-linear sum of active base-

elements at stage s as follows:

x = a0qσ0@ + · · ·+ aiqσi@ + b0qτ0@ + · · ·+ bjqτj@,

where bj = 0, σ0@ <lex · · · <lex σi@ <lex f <lex τ0@ <lex · · · <lex τj@.

(1) If there is a k ≤ i such that ak = 0, then find such a least k, and set the f -sign

of x to be the sign of ak.

(2) Otherwise, let k′ ≤ j be the least number with bk′ = 0, and set the f -sign of

x to be the sign of −bk′ .

Furthermore, if the expression of x changes later because some base-elements become

non-active, then the algorithm preserves the f -sign of x. This ensures that this sign

function is well-defined on G .

Clearly, the sign function is f -computable, so the corresponding order ≤f is f -

computable. Since there are σ@, τ@ ∈ 2<ω@ such that σ@ <lex f <lex τ@, qσ@ and

qτ@ have different sign under ≤f . Then ≤f is neither ≤G nor the reversal of ≤G ,

that is, ≤f is not computable.

We have constructed a computable group G such that for any set A, G admits

exactly two A-computable orders iff A is not PA-complete.


90

Chapter 5

Group orders and c.e. degrees

In this chapter, we will show that for any nonzero c.e. degree a, there is a nonzero

c.e. degree c < a and a computable group G such that G has orders of degree ≥ a

and c is group-order-computable via G , which implies that G has no incomputable

orders of degree ≤ c.

Theorem 5.1 For every noncomputable c.e. set A, there is a noncomputable c.e.

set C ≤T A and a computable torsion-free abelian group G with infinite rank such

that

(1) G has exactly two computable orders;

(2) G has an A-computable basis;

(3) every C-computable order on G is computable.

For a computable torsion-free abelian group A with infinite rank, if A has a

basis B, then the set O(A ) of degrees of orders on A contains d : d ≥ deg(B).Hence, our method is to control the degree of the constructed basis B := bi : i ∈ Nof G . If deg(B) ≤ a, then the constructed G also satisfies

d : d ≥ a ⊆ d : d ≥ deg(B) ⊆ O(G ).

5.1 Requirements and strategies

Let Hi : i ∈ N be a uniformly computable sequence of subgroups with rank one,

and let H = ⊕i∈N

Hi be the direct sum of Hi(i ∈ N). As in [3], we will build a

computable group G isomorphic to H which admits computable orders.

91

Chapter 5. Group orders and c.e. degrees

To prove Theorem 5.1, we will build a computable group G , a computable group

order ≤G on G , and a c.e. set C ≤T A to satisfy the following requirements.

Pe : We = C.

Q : G has an A-computable basis B = bi : i ∈ N.

Q0 : G ∼= H .

Q1 : ≤G and ≤∗G are exactly two computable orders on G .

Ne : If ≤Ce is a C-computable group order on G , then either ≤C

e =≤G or ≤Ce =≤∗

G .

Where We : e ∈ N is an enumeration of c.e. subsets of natural numbers, and

Φe : e ∈ N1 is an enumeration of partial computable functionals defined on G ×G ,

C is the complement of C. The symbol ≤Ce stands for the C-computable binary

relation on G computed by ΦCe ; specifically, we view for any x, y ∈ G , x ≤C

e y iff

ΦCe (x, y) ↓= 1. If ΦC

e (x, y) ↓, then let ϕCe (x, y) denote the use for the computation

ΦCe (x, y).

To ensure C ≤T A, we will add an A-permitting to the standard Pe-strategy.The Q-strategy ensures that G has an A-computable basis. The Q0-strategy and

the Q1-strategy are similar to corresponding strategies in Theorem 1.17. We will

develop new N -strategies.

5.1.1 A Pe-strategy with permitting

Pe is a standard Friedberg-Muchnik strategy with permitting. So we use cycles, we

start with cycle 0 and cycle k with k ≥ 0 proceeds as follows (assume y−1 = 0).

(1) Choose a big witness yk > yk−1.

(2) Wait for yk enumerating into We at some stage, say sk.

(3) Open cycle k + 1 and simultaneously, wait for Ask yk = As yk at some stage

s ≥ sk.

1Here, we explain Φe in more detail. We first fix an effective numbering, say ι, of G (i.e.,ι is a computably bijective mapping from N to G ). Let Θe : e ∈ N be an enumeration ofpartial computable functionals defined on N× N. Then for any X ⊆ N, and x, y ∈ G , ΦX

e (x, y) ≃ΘX

e (ι−1(x), ι−1(y)).

92


(4) Put yk into Cs, and close all cycles.

Suppose that the Pe-strategy opens infinitely many cycles. Then each cycle k is

waiting at (3) forever, that is, we find witnesses yk ∈ We and stages sk such that

Ask yk= A yk . Then we can compute A as follows: let x ∈ N, to decide whether it

is in A or not, find a least cycle kx such that ykx > x. Then x ∈ A⇔ x ∈ Askx .

Hence, by the noncomputability of A, there are only finitely many cycles opened

during the construction. If there is a first cycle k reaching at (4), then all cycles are

closed and yk ∈ C ∩We. Otherwise, no cycles are closed, and there is a least cycle k

waiting at (2) forever; in this case, cycle k is the largest cycle opened, yk /∈ We and

also yk /∈ C. In both cases, Pe is satisfied.We can compute C by A as follows: let x ∈ N, find a least stage sx such that

Asx x= A x (= y ∈ A : y ≤ x), then x ∈ C ⇔ x ∈ Csx .

5.1.2 The Q0-strategy

The Q0-strategy is a global strategy developed in [3]. To build the group G isomor-

phic to H , we maintain an approximation bs0, · · · , bss of a basis bi = limsbsi : i ∈ N

of G = ⊕i∈N

Gi and an approximation N s0 , · · · , N s

s of a basis Ni = limsN si : i ∈ N

of H = ⊕i∈N

Hi at each stage s such that

(1) bi and Ni is respectively a basis of rank one groups Gi and Hi;

(2) the isomorphism Ω : G = (G, 0G ,+G ) → H of abelian groups is determined

by sending each bi to Ni.

As each computable torsion-free abelian group of rank one can be effectively em-

bedded into Q, we view the subgroup Hi of H as a subgroup of additive group of

rational numbers. So N si ∈ Hi ⊆ Q and Ni ∈ Q.

Construction of G .

At stage 0. G0 = 0G , b00, where 0G is the zero element of G and b00 is the

approximation basis corresponding to N00 = 1.

At the end of stage s, Gs is the set containing all elements of G enumerated by

stage s, and each nonzero g ∈ Gs has assigned a unique Q-linear sum qs0bs0+· · ·+qsnbsn,

93


where qsn = 0(n ≤ s), each qsi (i ≤ n) in Q satisfying qsiNsi ∈ Hi at stage s. By

convention, 0G is assigned the empty sum.

At stage s+ 1.

Case 1. If no dependence relations are added among approximation basis ele-

ments by any Ne-strategy (e ≤ s), bs+1i = bsi and N

s+1i = N s

i for all i ≤ s. For each

g ∈ Gs, it has the same expression as before at stage s+1. That is, if qs0bs0+· · ·+qsnbsn

is the Q-linear sum of approximation basis elements at stage s, then let qs+1i = qsi

for all i ≤ n and g has the Q-linear sum qs+10 bs+1

0 + · · ·+ qs+1n bs+1

n at stage s+ 1.

Let N s+1s+1 = 1 and add a new element bs+1

s+1 into approximation basis. Find the

first tuple < x0, · · · , xm > of rationals (under a fixed effective numbering of the set

of all finite sequences of rationals) with xm = 0 and m ≤ s such that for all i ≤ m,

xiNs+1i ∈ Hi at stage s + 1, and the Q-linear sum x0b

s+10 + · · · + xmb

s+1m is not

assigned to elements of the constructed group. If such a tuple exist, then enumerate

x0bs+10 + · · ·+ xmb

s+1m into Gs+1.

Case 2. If some Ne-strategy adds a dependence relation of the form bsl = qbsk at

stage s with l an odd number, k an even number (e < k < l), q a rational number.

To ensure the uniqueness ofQ-linear sums at each stage, q also satisfies additional

conditions. We require that for any two elements in Gs, if their Q-linear sums of

approximation basis elements at stage s are different, then after substituting the

relation bsl = qbsk, their new Q-linear sums are still different.

We only need to consider Q-linear sums involving basis elements bsk and bsl .

• For example, assume that g = xbsk + ybsl , h = x′bsk + y′bsl , and either x = x′

or y = y′. If now after substituting bsl = qbsk, (x + yq)bsk = (x′ + y′q)bsk, i.e.,

x+ yq = x′ + y′q. Then y = y′ and q = x′−xy−y′ .

Thus, for any two different elements f, g in Gs, if after substituting the relation

bsl = qbsk, the new expression of f, g at stage s+1 is the same, then one can compute

a unique q(f, g) such that q = q(f, g). As Gs is a finite set, there are finitely many

such q(f, g). We will choose the desired q different from all q(f, g) with f = g in Gs

but f, g has the same expression at stage s + 1 when adding bsl = q(f, g)bsk. Then

for any two different elements in Gs, their assigned sums remain different at stage

s+ 1.

94


Now we update basis approximation elements. Let bs+1i = bsi for all i ≤ s with

i = l, and N s+1i = N s

i for all i ≤ s with i = k. Add two new elements bs+1l and bs+1

s+1

into approximation basis. Set N s+1s+1 = 1. We now define N s+1

k .

Let g ∈ Gs.

• Assume that g = qs0bs0 + · · ·+ qskb

sk + · · ·+ qsl b

sl + · · ·+ qsnb

sn at stage s. Now g

changes to

qs0bs0 + · · ·+ (qsk + qsl q)b

sk + · · ·+ 0bsl + · · ·+ qsnb

sn.

Let qs+1i = qsi for all i ≤ n with i /∈ k, l, qs+1

k = qsk + qsl q, qs+1l = 0. Then at

stage s+ 1, g = qs+10 bs+1

0 + · · ·+ qs+1n bs+1

n .

To ensure the Q-linear sum of g is well-defined after substituting the dependence

relation bsl = qbsk, we need qs+1i N s+1

i ∈ Hi for all i ≤ n. For i /∈ k, l, qs+1i N s+1

i =

qsiNsi ∈ Hi; note that qs+1

l N s+1l = 0 ∈ Hl is already true. So we only require

qs+1k N s+1

k ∈ Hk.

Now define N s+1k = dqdN

sk , where dq is the denominator of q and d is the product

of all dqsl such that dqsl is the denominator of nonzero qsl , and qsl bsl appears in the

Q-linear sum of an element in Gs. Then qdq ∈ Z and qsl d ∈ Z, so

qs+1k N s+1

k = (qsk + qsl q)Ns+1k = dqdq

skN

sk + qsl dqdqN

sk ∈ Hk.

Finally, search the first tuple < x0, · · · , xm > and add the Q-linear sum x0bs+10 +

· · ·+ xmbs+1m into Gs+1 as in case 1.

Case 3. If some Ne-strategy adds a dependence relation of the form bsl =

m1bsj −m2b

sk at stage s with l an odd number, l > j, k an even number (e < k < l),

m1,m2 integers.

The first condition of m1,m2 is to keep the assignment of sums of elements in

Gs remaining one-to-one after substituting the relation bsl = m1bsj −m2b

sk.

• For example, assume that g = xbsj+ybsk+zb

sl , h = x′bsj+y

′bsk+z′bsl , and g = h

in Gs, that is, x = x′ or y = y′ or z = z′. If

(x+ zm1)bsj + (y − zm2)b

sk = (x′ + z′m1)b

sj + (y′ − z′m2)b

sk,

then x+ zm1 = x′+ z′m1 and y− zm2 = y′− z′m2. So z = z′, m1 =x′−xz−z′ , and

m2 =y−y′z−z′ .

95


In general, for f = g in Gs, so their expressions as Q-linear sum of approximation

basis elements are different at stage s. If f, g has the same expression at stage s+1

after substituting the relation bsl = m1bsj −m2b

sk, then for i = 1, 2, we can compute

a unique mi(f, g) such that mi = mi(f, g). Note that there are finitely many such

mi(f, g), the desired mi will be different from all those mi(f, g) with f = g in Gs.

Then the Q-linear sum of elements in Gs remains one-to-one at stage s+ 1.

Let bs+1i = bsi for all i ≤ s with i = l and N s+1

i = N si for all i ≤ s. Add two new

elements bs+1l and bs+1

s+1 into approximation basis, and set N s+1s+1 = 1.

The second conditions on m1,m2 are to ensure that the Q-linear sums are well-

defined after substituting the dependence relation bsl = m1bsj − m2b

sk. That is, for

any g ∈ Gs,

• assume that g = qs0bs0 + · · · + qsjb

sj + · · · + qskb

sk + · · · + qsl b

sl + · · · + qsnb

sn. Now

rewrite g as

qs0bs0 + · · ·+ (qsj + qslm1)b

sj + · · ·+ (qsk − qslm2)b

sk + · · ·+ 0bsl + · · ·+ qsnb

sn.

Let qs+1i = qsi for all i ≤ n with i /∈ j, k, l, qs+1

j = qsj+qslm1, q

s+1k = qsk−qslm2,

qs+1l = 0. Then

g = qs+10 bs+1

0 + · · ·+ qs+1n bs+1

n .

We need to ensure qs+1i N s+1

i ∈ Hi for all i ≤ n.

Note that qs+1i N s+1

i ∈ Hi for i /∈ k, j, l and qs+1l N s+1

l = 0 ∈ Hl. We only

require qs+1j N s+1

j ∈ Hj and qs+1k N s+1

k ∈ Hk.

qs+1j N s+1

j = (qsj + qslm1)Ns+1j = qsjN

sj + qslm1N

sj .

Choose m1 to be the multiple of the product d of all dqsl such that dqsl is the denom-

inator of qsl , where qsl = 0 and qsl b

sl appears in the Q-linear sum of an element in Gs.

Then qslm1 ∈ Z and qs+1j N s+1

j ∈ Hj. Similarly, choose m2 with d|m2, we have

qs+1k N s+1

k = (qsk − qslm2)Ns+1k = qskN

sk − qslm2N

sk ∈ Hk.

At last, find the first tuple < x0, · · · , xm > of rational numbers and add the

Q-linear sum x0bs+10 + · · ·+ xmb

s+1m into Gs+1 as in case 1.

96


For f, g, h ∈ Gs, define f +s g = h if the addition of Q-linear sums of f and g is

equal to the Q-linear sum of h. By linearity of additions, when updating expressions

of f, g, h at stage s+ 1, we still have f +s+1 g = h.

This completes the construction of G at stage s+ 1.

Let Ωs : Gs → H be the linear extension of mapping which sends each bsi to Nsi .

Then Ω = limsΩs is a ∆0

2-isomorphism from G to H such that Ω(bi) = Ni for all i.

5.1.3 The Q1-strategy

TheQ1-strategy is also a global strategy developed in [3]. Similar to theQ0-strategy,

at each stage s, we maintain approximation elements rs0, · · · , rss in R which is

linearly independent over Q, and map bsi to rsi . We choose each rsi (1 ≤ i ≤ s) as a

positive rational multiple of√pi, where pi is the i-th prime number for i ≥ 1.

For i ≥ 1, ri = limsrsi . r0 = rs0 = 1R for all s.

(1) Let Θ be the linear extension of the map Ψ : G → (R, 0R,+R); bi 7→ ri;

(2) Let Θ(G ) be the image of G under Θ, then G is isomorphic to the subgroup

Θ(G ) of (R, 0R,+R) and ri : i ∈ N is a basis of Θ(G ).

The linear order ≤G will satisfy x ≤G y ⇔ Θ(x) ≤R Θ(y) for any x, y ∈ G , this

definition does not ensure the computability of ≤G . To make ≤G computable, we

will approximate ≤G via ≤s at each stage s, and keep x ≤s y implies x ≤t y for all

stages t ≥ s.

N si ∈ Hi ⊆ Q

bsi ∈ Gi

Ωs

88qqqqqqqqqqq

Θs

&&NNNNNNNNNN

rsi ∈ R

At each stage s, each approximation basis bsi (i ≤ s) will be mapped to rsi in R,

and we have an approximation Θs of Θ which is a linear extension of mapping on

approximation basis elements. Θ = limsΘs is a ∆0

2 map from G to R. Although at

97


limit level, the linear order ≤G is defined by Θ which is a linear extension on basis

elements of G , we cannot simply define the approximation ≤s of ≤G at stage s via

Θs as: g ≤s h⇔ Θs(g) ≤R Θs(h).

The obstacle is when some Ne-strategy adding dependence relations of the form

bsl = qbsk or bsl = m1bsj −m2b

sk at stage s+ 1, if ≤s+1 is really defined by Θs+1, note

Θs(bsl ) = rsl , this means that bsl maps to new positions Θs+1(b

sl ) = qrsk or Θs+1(b

sl ) =

m1rsj − m2r

sk in R at stage s + 1 (we will see rs+1

k = rsk and rs+1j = rsj). We may

change the relative positions of two elements in Gs, that is, maybe Θs(x) <R Θs(y)

but Θs+1(y) <R Θs+1(x). So we cannot ensure that ≤G is computable.

The method used in [3] is to assign rational-endpoint interval (asi , asi ) to each basis

element bsi to control the position rsi of bsi , namely, let rsi ∈ (asi , a

si ) and a

si−asi <R

12s,

and then control the position of each g ∈ Gs in R by assigning a rational-endpoint

interval (asg, asg) for g based on the intervals of basis elements. That is, if g =

x0bs0 + · · ·+ xnb

sn, then

(i) asg = x0σ(as0)+ · · ·+xnσ(a

sn) with σ(a

si ) = asi if xi ≥R 0, σ(asi ) = asi if xi <R 0;

(ii) asg = x0τ(as0) + · · ·+ xnτ(a

sn) with τ(a

si ) = asi if xi ≥R 0, τ(asi ) = asi if xi <R 0.

Now define ≤s via assigned intervals as follows.

(1) Let g ≤s g for each g ∈ Gs.

(2) For each g, h ∈ Gs, if (asg, a

sg) ∩ (ash, a

sh) = ∅ and asg <R a

sh, then g <s h.

In order to make ≤G computable, we choose approximation intervals such that

rs+1i ∈ (as+1

i , as+1i ) ⊆ (asi , a

si )

and as+1i − as+1

i <R1

2s+1 . Then for each g ∈ Gs, (as+1g , as+1

g ) ⊂ (asg, asg). Hence, for

all g, h ∈ Gs, g <s h implies g <s+1 h.

Note that bsk never changes for even k, as in [3], we also require that rsk close to

0R and choose ask <R12s

for positive even k. This helps us to deal with witnesses for

N -strategies.

Construction of ≤G .

98


At stage 0. G0 = 0G , b00, 0G sends to 0R, and set 0G <0 b

00 by defining r0 = 1R.

Assume that we have defined rs0, · · · , rss by stage s.

At stage s+ 1.

Case 1. If no dependence relations are added via any Ne-strategy. For each

i ≤ s, rs+1i = rsi . Choose rational numbers as+1

i , as+1i such that rs+1

i ∈ (as+1i , as+1

i ) ⊆(asi , a

si ) and a

s+1i − as+1

i <R1

2s+1 . If i is even, we also let as+1i <R

12s+1 .

Add a new element of the form rs+1s+1 = qs+1

√ps+1 with qs+1 a positive rational

number. Choose as+1s+1, a

s+1s+1 such that rs+1

s+1 ∈ (as+1s+1, a

s+1s+1) and a

s+1s+1 − as+1

s+1 <R1

2s+1 . If

s+ 1 is even, we also require as+1s+1 <R

12s+1 .

Case 2. If some Ne-strategy adds a dependence relation of the form bsl = qbsk

with q ∈ Q+ or bsl = m1bsj −m2b

sk with m1,m2 ∈ Z. For each i ≤ s with i = l, set

rs+1i = rsi and assign (as+1

i , as+1i ) as in case 1. For bsl ,

(1) if bsl = qbsk, choose q such that the new interval (qas+1k , qas+1

k ) for bsl at stage

s+ 1 is contained in (asl , asl );

(2) if bsl = m1bsj − m2b

sk with m1,m2 ∈ Z+, choose m1,m2 such that the new

interval (m1as+1j −m2a

s+1k ,m1a

s+1j −m2a

s+1k ) for bsl at stage s+1 is contained

in (asl , asl );

(3) if bsl = m1bsk − m2b

sj with m1,m2 ∈ Z+, choose m1,m2 such that the new

interval (m1as+1k −m2a

s+1j ,m1a

s+1k −m2a

s+1j ) for bsl at stage s+1 is contained

in (asl , asl ).

Let rs+1l =

√pl, corresponding to new bs+1

l . Choose new rational numbers

as+1l , as+1

l such that rs+1l ∈ (as+1

l , as+1l ) and as+1

l − as+1l <R

12s+1 . Choose a new

rs+1s+1 for bs+1

s+1 and assign interval (as+1s+1, a

s+1s+1) as in case 1.

This completes the construction of ≤G at stage s+ 1.

5.1.4 The Q-strategy

In order to satisfy the Q-requirement, we modify the Ne-strategy developed in [3]

through the following way: we add dependence relations of the form bsl = qbsk or

99


bsl = mbsk + nbsj for some odd l at stage w only when permitted by A, that is,

Aw−1 l = Aw l. Then for any stage t, At l= A l implies bl = btl .

The main idea is that for same active witness of Ne, we may search for multiple

diagonalization witnesses until permitted by A, and then add dependence relations

[here, active witness and diagonalization witness will be defined in the basic Ne-

strategy]. As A is assumed to be incomputable, such a permitting occurs; otherwise,

we can initiate an effective process for computing A.

Hence, B = bi = limsbsi : i ∈ N ≤T A. Indeed, for an arbitrary g ∈ G , let sg

be the first stage when g enumerates into G . If g is neither bsgsg nor b

sgl with odd

l < sg, then g /∈ B. Otherwise, g = bsgsg or g = b

sgl for some odd l (in this case, we

add dependence relations on bsg−1l , and b

sgl is the new approximation for bl),

(1) if g = bsgsg with sg even or g = b

sgl for some odd l < sg, then g ∈ B;

(2) if g = bsgsg with sg odd, find a least stage t ≥ sg such that At sg= A sg , then

g ∈ B iff g = btsg = bsg .

Ne still acts finitely often, so the whole construction is still a finite injury argu-

ment.

5.1.5 A Ne-strategy

We need to ensure that if ≤Ce =≤G and ≤C

e =≤∗G , then ≤C

e is not a group order on

G . A basic Ne-strategy works as follows.

(A1) Wait for an active witness (bsj , n, bsk) at some stage s, namely, (bsj , n, b

sk) satis-

fying (a1), (a2), (a3) or (a1), (a2), (a4).

(a1) k is an even number larger than e.

(a2) 0G <s nbsk <s b

sj <s (n+ 1)bsk.

(a3) bsk >Ce 0G and either bsj <

Ce nb

sk or bsj >

Ce (n+ 1)bsk.

(a4) bsk>Ce∗0G and either bsj<

Ce∗nbsk or bsj>

Ce∗(n+ 1)bsk.

By Lemma 3.9 [3], as long as ≤Ce is a group order on G satisfying ≤C

e =≤G

and ≤Ce =≤∗

G , we can find active witnesses for ≤Ce . That is, if no active witnesses

appeared during the construction, then either ≤Ce is not a group order on G or

≤Ce =≤G or ≤C

e =≤∗G , and hence Ne is satisfied.

100


(A2) We first set a restraint rC2 (bsj , n, b

sk) on C to preserve the computations or-

derings in (a3) or (a4) for ≤Ce at stage s. Say that Ne has active witnesses

(bsj , n, bsk) now.

For the active witness (bsj , n, bsk), assume that we are in case (a3), we deal with (a4)

similarly by considering the reveral order ≤Ce∗.

As long as (bsj , n, bsk) is not initialized, that is, the C-restraint rC2 (b

sj , n, b

sk) still

exist and bsj is not canceled, we use the mechanism of cycles to assign multiple

diagonalization witnesses for (bsj , n, bsk). Cycles can be opened or closed, and cycle

i can open cycle i + 1. Assume that we have opened cycles n < i. During cycle i

with i ≥ 0, proceed as follows:

(D1) At the least odd stage ti > ti−1 ≥ s (set t−1 = s), declare that the new

approximate basis element btiti is a diagonalization witness for (bsj , n, bsk) at

cycle i. Order btiti as in (b1) or (b2) when (b0) holds.

(b0) For all stages v with s ≤ v ≤ ti, bvj = bsj , Cs rC2 (bsj ,n,b

sk)= Cti rC2 (bsj ,n,b

sk).

(b1) If we have bsj <Ce nb

sk in (a3), set nbsk <ti b

titi <ti b

sj , that is,

• define rtiti = qti√pti for some positive rational qti and choose atiti , a

titi such

that natik <R atiti <R r

titi <R a

titi <R a

tij .

(b2) If we have bsj >Ce (n+ 1)bsk in (a3), set bsj <ti b

titi <ti (n+ 1)bsk, that is,

• define rtiti = qti√pti for some positive rational qti and choose atiti , a

titi such

that atij <R atiti <R r

titi <R a

titi <R (n+ 1)atik .

We call (btiti , bsj , n, b

sk) a pre-potentially permanent witness at cycle i of the active

witnesses (bsj , n, bsk) of Ne. Proceed as follows to add dependence relations.

(D2) Wait for a potentially permanent witness (buti , bsj , n, b

sk) at some stage u > ti,

namely, (buti , bsj , n, b

sk) satisfying (c1), (c2) or (c1), (c3) or (c1), (c4).

(c1) For all stages v with ti ≤ v ≤ u, bvti = btiti , for all stages v with s ≤ v ≤ u,

bvj = bsj , and Cs rC2 (bsj ,n,bsk)= Cu rC2 (bsj ,n,b

sk).

101


(c2) buti <Ce nb

sk.

(c3) nbsk <Ce b

uti<Ce (n+ 1)bsk.

(c4) (n+ 1)bsk <Ce b

uti.

When (c1) holds, if (c2) or (c3) or (c4) never appears, then ≤Ce does not satisfy

totality of linear orders, we have done.

(D3) We first set a restraint rC4 (buti, n, bsk) ≥ rC2 (b

sj , n, b

sk) on C to preserve ≤C

e -

orderings in (c2) or (c3) or (c4). Now Ne has a potentially permanent witness

(buti , bsj , n, b

sk). Open cycle i + 1 to look for new diagonalization witnesses for

the active witness (bsj , n, bsk).

(D4) Wait for A ti to change.

(D5) Suppose that A ti changes at some least stage w ≥ u > ti. That is, w is the

least stage ≥ u at which some element x ≤ ti enumerates into A. If (d0) holds,

stop all cycles for the active witnesses (bsj , n, bsk), and proceed as in (D5.1) or

(D5.2) to add dependence relations.

(d0) For all stage v with ti ≤ v ≤ w−1, bvti = btiti , for all stage v with s ≤ v ≤ w,

bvj = bsj , and Cu rC4 (buti,n,bsk)

= Cw rC4 (buti,n,bsk)

.

(D5.1) If bw−1ti <C

e nbsk or (n + 1)bsk <Ce bw−1

ti , then add dependence relations of the

form bw−1ti = qbwk , where q is properly chosen to diagonalize group orders and q

also satisfies the conditions asserted by the Q0-strategy and the Q1-strategy.

We now state the conditions of q.

(d1) q ∈ (n, n + 1). Then if ≤Ce is a group order on G , as q ∈ (n, n + 1) and

bk >Ce 0G , we have nbk ≤C

e bw−1ti = qbk ≤C

e (n + 1)bk, contradicting with

the assumption for ≤Ce -orderings on b

w−1ti , nbsk and (n+ 1)bsk.

(d2) If bw−1ti = qbwk , we need to find a rational-endpoint interval (awk , a

wk ) for b

wk

under the Q1-strategy. Then we need to choose q such that (qawk , qawk ) ⊆

(aw−1ti , aw−1

ti ). As nbw−1k <w−1 b

w−1ti <w−1 (n+1)bw−1

k , this means nrw−1k <R

aw−1ti <R a

w−1ti <R (n + 1)rw−1

k . Hence, we can find infinitely many q in

(n, n + 1) with aw−1ti <R qrw−1

k <R aw−1ti , and for each such a q, we can

choose proper awk , awk satisfying the conditions in the Q1-strategy.

102


(d3) q satisfies the conditions under the Q0-strategy, that is, q = q(f, g) for

any f, g ∈ Gw−1, such a q(f, g) turns an inequality f = g in Gw−1 to be

an equality f = g in Gw when adding bw−1ti = q(f, g)bwk . Note that there

are only finitely many such q(f, g) as Gw−1 is a finite set.

(d4) Now we have showed that there are infinitely many q satisfying (d1), (d2),

(d3). Our final q will be the least one of them such that the numbering

of q (we fix an effective numbering of Q) is larger than the current stage

w.

(D5.2) If nbsk <Ce bw−1

ti <Ce (n + 1)bsk, then we add dependence relations according to

whether bw−1ti <w−1 b

w−1j [i.e., we are in (b1)], or bw−1

ti >w−1 bw−1j [i.e., we are in

(b2)]. Remember that our target is still to diagonalize axioms of group orders

and to make the diagonalization to be compatible with the Q0-strategy and

the Q1-strategy.

(D5.2.1) If bw−1ti <w−1 b

w−1j . Our environment for ≤G and ≤C

e is

≤G : nbw−1k <w−1 b

w−1ti

<w−1 bw−1j <w−1 (n+ 1)bw−1

k

≤Ce : bw−1

j <Ce nb

w−1k <C

e bw−1ti

<Ce (n+ 1)bw−1

k

Add dependence relations of the form bw−1ti = m1b

wj − m2b

wk with m1 >N 1.

On the one hand, in order to be compatible with ≤G , (note that bwj = bw−1j ,

bwk = bw−1k ), we need

nbw−1k <w−1 m1b

w−1j −m2b

w−1k <w−1 b

w−1j <w−1 (n+ 1)bw−1

k

Note that bw−1k >G 0G . Then

(m1 − 1)nbw−1k <G (m1 − 1)bw−1

j <G m2bw−1k

and thus (m1 − 1)n <N m2.

On the other hand, if we want the dependence relation to be compatible with

≤Ce , then similarly, we have

bw−1j <C

e nbw−1k <C

e m1bw−1j −m2b

w−1k <C

e (n+ 1)bw−1k

103


Note that bw−1k >C

e 0G . Then

m2bw−1k <C

e (m1 − 1)bw−1j <C

e (m1 − 1)nbw−1k

and thus m2 <N (m1 − 1)n.

Before looking for additional conditions on m1,m2, we recall the following

important lemma which will be used.

Lemma 5.1 [3] If r1, r2 ⊆ R+ is linearly independent over Q, then for any

two rational numbers q1 < q2, and any integer n ≥ 1, there are infinitely many

positive integers m1,m2 such that n|m1, n|m2 and m1r1 −m2r2 ∈ (q1, q2).

The desired m1,m2 ∈ N+ will satisfy the following conditions:

(f1) Choose m1,m2 ∈ N such that nd|m1, nd|m2 and 1 <N m1 ≤Nm2

n, where

d is the product of all denominators of nonzero qw−1ti with qw−1

ti bw−1ti ap-

pearing in the Q-linear sum of some element of Gw−1. Then ≤Ce cannot

be a group order on G .

(f2) If bw−1ti = m1b

wj −m2b

wk , under the Q1-strategy, we need to find awk , a

wk ,

awj and awj such that

rwk = rw−1k ∈ (awk , a

wk ) ⊆ (aw−1

k , aw−1k ), rwj = rw−1

j ∈ (awj , awj ) ⊆ (aw−1

j , aw−1j )

where awk − awk <R12w, awj − awj <R

12w

and

(♣) m1rwj −m2r

wk ∈ (m1a

wj −m2a

wk ,m1a

wj −m2a

wk ) ⊆ (aw−1

ti, aw−1

ti)

By Lemma 5.1, there are infinitely many natural numbers m1, m2 with

nd|m1, nd|m2 such that m1rwj − m2r

wk ∈ (aw−1

ti , aw−1ti ). Then for such

m1,m2, we can find proper awk , awk , a

wj and awj satisfying (♣). Moreover,

the calculation above shows that m1 ≤Nm2

n.

(f3) To make sure the Q-linear sums of elements in Gw−1 remaining one-to-

one, we also require that for i = 1, 2, mi = mi(f, g) for any f, g ∈ Gw−1,

where adding bw−1ti = m1(f, g)b

wj −m2(f, g)b

wk entails f = g at stage w,

but f = g at stage w − 1. There are only finitely many such mi(f, g), so

we can find infinitely many desired m1,m2 satisfying (f1), (f2), (f3).

104


(f4) The finalm1,m2 will satisfy (f1), (f2), (f3) withm1,m2 > w and ⟨m1,m2⟩least.

(D5.2.2) If bw−1ti >w−1 b

w−1j . Our environment for ≤G and ≤C

e is

≤G : nbw−1k <w−1 b

w−1j <w−1 b

w−1ti

<w−1 (n+ 1)bw−1k

≤Ce : nbw−1

k <Ce b

w−1ti

<Ce (n+ 1)bw−1

k <Ce b

w−1j

Add dependence relations of the form bw−1ti = m1b

wk −m2b

wj . We choose m1,m2

such that substituting the relation into ≤G holds, that is,

nbw−1k <w−1 b

w−1j <w−1 m1b

w−1k −m2b

w−1j <w−1 (n+ 1)bw−1

k

Then

(m2 + 1)[m1 − (n+ 1)]bw−1k <w−1 (m2 + 1)m2b

w−1j

<w−1 m2m1bw−1k

So (m2 + 1)[m1 − (n+ 1)] <N m2m1, and thus m1 <N (n+ 1)(m2 + 1).

Now under the conditions m1 <N (n + 1)(m2 + 1) and (n + 1)|m1,m1

(n+1)<N

m2 + 1, that is, m1

(n+1)≤N m2. Then ≤C

e is not a group order on G . Indeed,

suppose that ≤Ce is a group order on G , then

0G <Ce nb

w−1k <C

e bw−1ti

= m1bw−1k −m2b

w−1j

≤Ce m1b

w−1k − m1

n+ 1bw−1j

=m1

n+ 1((n+ 1)bw−1

k − bw−1j ) <C

e 0G

We now determine m1,m2:

(g1) Choose m1,m2 ∈ N such that (n+1)d|m1, (n+1)d|m2 andm1

(n+1)≤N m2,

where d is the same as in (f1).

(g2) Assume that bw−1ti = m1b

wk −m2b

wj . We need to assign rational-endpoint

intervals for bw−1ti at stage w, that is, find awk , a

wk , a

wj and awj the same as

in (f2) but with the condition (♣) changed to

(♠) m1rwk −m2r

wj ∈ (m1a

wk −m2a

wj ,m1a

wk −m2a

wj ) ⊆ (aw−1

ti, aw−1

ti)

105


According to Lemma 5.1, there are infinitely many natural numbers

m1,m2 with (n+1)d|m1, (n+1)d|m2 such thatm1rwk−m2r

wj ∈ (aw−1

ti , aw−1ti ).

Then we can choose proper (awk , awk ) ⊆ (aw−1

k , aw−1k ), and proper (awj , a

wj ) ⊆

(aw−1j , aw−1

j ) satisfying (♠). Then m1

(n+1)≤N m2 automatically holds.

(g3) The same as in (f3), we require that the choice of m1 and m2 preserves

the one-to-one assignment of Q-linear sums to elements in Gw−1.

(g4) The final m1 and m2 satisfy (g1), (g2), (g3), m1 > w, m2 > w with

⟨m1,m2⟩ least.

This ends the description of the basic Ne-strategy.

The interactions between two N -strategies are finite as in [3]. They both set

C-restraints, so there is no confliction in building C; the confliction is in choosing

witnesses.

Suppose that Ne choose an active witness (bsj , n, bsk) at stage s, and then add a

diagonalization witness btt at an odd stage t > s. To diagonalize axioms of group

orders, Ne may add dependence relations to change btt at stages > t. However, some

Ni may find an active witness (buj′ , n′, buk′) at stage u > t with buj′ = btt, and then add

a diagonalization witness bww at an odd stage w > u. To preserve its active witness,

Ni requires btt keeping the same at stages > u.

The solution is a priority argument. When e < i,

(1) if Ne acts first to add dependence relations, it changes btt to initialize Ni’s

active witnesses (Ne is satisfied and cannot injure Ni later, Ni will choose new

active witnesses at later stages);

(2) if Ni acts first, it adds dependence relations on bww and hence does not change

buj′(= btt), so no confliction occurs.

When i < e,

• if Ni finds its active witness, say (buj′ , n′, buk′), at stage u > t with buj′ = btt, it

cancels Ne’s diagonalization witness btt to avoid potential conflictions (Ne will

find its diagonalization witness bt′

t′ with t′ big at later odd stages).

106


P-requirements are purely set-theoretic requirements to ensure the noncom-

putability of C. Each Pe is positive on C and acts at most once to enumerate

a permitted witness y ∈ We into C. So Pe injures negative requirements with lower

priority which may set restraints on C finitely often.

5.2 Construction

Q, Q0 and Q1 are global requirements. The priority of other requirements is as

follows:

N0 ≺ P0 ≺ · · ·Ne ≺ Pe ≺ · · · .

The construction mainly splits into two parts. One is set-theoretic construc-

tions, that is, building set C. The other is algebraic constructions, that is, adding

dependence relations to defeat ≤Ce to be a group order on G , and then constructing

G and ≤G according to whether some dependence relations are added or not.

Construction.

Stage 0. Let C0 = ∅. Build G0 and ≤0 by the Q0-strategy in Section 5.1.2 and

the Q1-strategy in Section 5.1.3 respectively.

Stage s > 0.

Step 1. Find the highest priority requirement, say Q, among

N0 ≺ P0 ≺ · · ·Ns ≺ Ps

which requires attention (such a requirement exists since Ps requires attention at

stage s). Act correspondingly and initialize all strategies ≻ Q. Goto Step 2 if no

dependence relations are added; goto Step 3 otherwise.

Say that Pe is satisfied at stage s if We,s ∩ Cs = ∅, where We,s is the set of all

elements enumerated into We by stage s. Say that Pe requires attention at stage s

if it is unsatisfied and one of the following (P1) and (P2) happens. Act accordingly.

(P1) Pe has no cycles opened at stage s.

Action: Open cycle 0 by appointing a large witness y0, and start to wait for

y0 enumerating into We. Proceed as in the basic Pe-strategy.

107


(P2) Pe has a previous witnesses yk ∈ We,sk , cycle k + 1 is opened at stage sk

and cycle k started to wait for Ask yk-changes at stage sk. Now Ask yk=As−1 yk = As yk and stage s ≥ sk is the first stage at which some cycles of Pereach (4) of the basic Pe-strategy, and this cycle is just cycle k.

Action: Put yk into Cs, close all opened cycles of Pe. Declare that Pe is

satisfied.

Say that Ne is satisfied if we add dependence relations between some active

witness and some diagonalization witness. That is, we have finished the following

actions.

(1) We find some active witness, say (bsj , n, bsk), and assign C-restraints to protect

this active witness.

(2) We arrive at some cycle i for (bsj , n, bsk) with some diagonalization witness btiti

and set C-restraints to protect potentially permanent witness (buti , bsj , n, b

sk) at

some stage u ≥ ti.

(3) A ti changes after stage u, and we add dependence relations between diago-

nalization witness and active witness.

In this case, ≤Ce is not an order on G .

Say that Ne requires attention at stage s if it is not satisfied at stage s, and one

of the following holds at stage s. Act correspondingly.

(N1) Some active witness appears at stage s.

Action: First, set C-restraints to protect this active witness. Second, if some

Ni-strategy (i > e) has a diagonalization witness in some cycle, cycle j say, of

some active witness which occurs in the active witness of Ne, then cancel the

diagonalization witness at cycle k, where k ≥ j, of Ni’s active witness.

(N2) Ne has an active witness and no cycles opened for this active witness at stage

s.

Action: Open cycle 0 of the active witness to find diagonalization witnesses.

108


(N3) Ne has some active witness, and some cycle i is opened for this active witness.

No diagonalization witness is found in cycle i at stage s, and s is odd.

Action: Set the diagonalization witness btiti = bss at cycle i of the active witness,

and order bss with the active witness in (b1) or (b2) as in the basic Ne-strategy.

(N4) Ne has some active witness, and some cycle i with uncanceled diagonalization

witness btiti is opened for this active witness. Now one of the ordering relations

between the active witness and btiti in (c2) or (c3) or (c4) appears at stage s.

Action: Set C-restraints to protect the potentially permanent witnesses at

cycle i. Open cycle i+1 for the active witness to seek for new diagonalization

witnesses and simultaneously, wait for A ti to change.

(N5) Ne has a potentially permanent witnesses with diagonalization witness btiti in

cycle i, and some x ≤ ti first enumerates into A at stage s.

Action: For such a least cycle i, add dependence relations on btiti according

to the basic Ne-strategy, and then stop all cycles of Ne. Declare that Ne is

satisfied.

Step 2. Build Gs and ≤s respectively according to the Q0-strategy in Section

5.1.2 and the Q1-strategy in Section 5.1.3 with no dependence relations added.

Step 3. Build Gs and ≤s respectively according to the Q0-strategy in Section

5.1.2 and the Q1-strategy in Section 5.1.3 with the added dependence relation.

This ends the stage s of the construction.

5.3 Verification


(1) Q can be initialized at most finitely often;


(3) Q initializes strategies with lower priority finitely often.

109


Proof: We will show that for each e, (1)-(3) hold for all Ne,Pe. We will prove the

lemma by induction. For e = 0. N0 has the highest priority, it cannot be initialized.

(1) holds. We now show that it acts finitely often and is satisfied.

If ≤C0 has no active witnesses during the construction, then by Lemma 3.9 in [3],

either ≤C0 is not a group order, or ≤C

0 =≤G or ≤C0 =≤∗

G . N0 is satisfied.

Suppose that there is some active witness (bsj , n, bsk) for N0 at stage s. Then N0

acts to protect (bsj , n, bsk) by setting C-restraints, and initializes all strategies with

lower priority.

Under the basic N0-strategy, it starts by opening cycle 0, that is, by waiting for

a diagonalization witness bt0t0 at some odd stage t0 > s and then waiting for

(♠0) : bt0t0 <C0 nb

sk ∨ nbsk <C

0 bt0t0 <

C0 (n+ 1)bsk ∨ (n+ 1)bsk <

C0 b

t0t0 .

In general, cycle i > 0 works as follows.

(i) Wait for a diagonalization witness btiti at some odd stage ti > ui−1 > ti−1 and

then wait for

(♠i) : btiti <C0 nb

sk ∨ nbsk <C

0 btiti <

C0 (n+ 1)bsk ∨ (n+ 1)bsk <

C0 b

titi .

If (♠i) never happens, then N0 is satisfied because ≤C0 does not satisfy the

axioms of linear orders.

(ii) If (♠i) happens at some stage ui > ti. Open cycle i + 1 and simultaneously

wait for A ti to change.

– If A ti changes at some stage wi > ui, add dependence relations between

btiti and (bsj , n, bsk). Close all cycles of (bsj , n, b

sk), and N0 is satisfied.

If there is a cycle i which waits at (♠i) forever, then cycle i+1 is never opened.

In this case, ≤C0 is not a linear order and thus N0 is satisfied. So assume that no

cycles i wait at (♠i) forever.

Suppose that each cycle i is not closed, then each cycle i waits for A ti to change

forever, that is, Aui ti= A ti . Note that s ≤ ti < ui < ti+1 for all i. Now we can

compute A as follows: for any x, find the least ti with x < ti, then x ∈ A iff x ∈ Aui .

110


As A is assumed to be noncomputable, there is a cycle i which provides a diag-

onalization witness btiti , and Aui ti changes later, say at stage t′ ≥ ui. Then we add

dependence relations between btiti and (bsj , n, bsk) at stage t

′, and N0 is satisfied.

Hence, in all cases above, N0 acts finitely many times and is satisfied at end. (2)

and (3) hold.

Let s0 be the least stage such that N0 never acts later, then P0 is not injured

after stage s0. Let t0 ≥ s0 be the least stage such that cycle 0 of P0 is opened. As

long as P0 is not satisfied, the basic P0-strategy will open cycles, say k, by picking

a new witness yk at cycle k and starting to wait for yk ∈ W0; when yk ∈ W0, yk is

enumerated into C at some stage s only if As yk = As−1 yk . As A is noncomputable,

if such witnesses eventually enumerate into W0, then this A-permitting happens at

certain cycles. So P0-strategy can only open finitely many cycles. That is, P0 acts

finitely often after stage s0.

As long as P0 is not satisfied automatically, there is a cycle k such that either yk

is never put into C in which case yk /∈ W0 or yk is put into C in which case yk ∈ W0.

In both cases, W0 = C. So P0 is satisfied after finitely many actions, thus (2) and

(3) hold.

Now assume that the lemma holds for all Ni,Pi with i < e. By induction

hypothesis, there is a stage se such that Ni,Pi with i < e are satisfied at stage se.

Then Ne has the highest priority after stage se. Similar to case e = 0, (1)-(3) hold

for Ne and Pe.

Lemma 5.3 C ≤T A.

Proof: During the construction, only permitted elements can be enumerated into

C. That is, x ∈ Cs only if there is some small element z ≤ x such that z ∈ As−As−1.

To decide whether x ∈ C or not, by using A, find a least stage s such that for all

z ≤ x, z ∈ A⇔ z ∈ As. Then x ∈ C ⇔ x ∈ Cs.

The differences between Theorem 5.1 and Theorem 1.17 are that elements enu-

merated into C are permitted by A and that the diagonalization witnesses are per-

mitted by A. We now show that for each i, the approximation basis element bsi only

changes finitely often. Then limsbsi exists, and thus we can satisfy Q0 and Q1 the

same as in Theorem 1.17.

111


Lemma 5.4 For each i, bi = limsbsi exists.

Proof: For each i, consider bii. It may change at later stages only if bii is chosen

as a diagonalization witness in some cycle of some active witnesses for some N -

requirements. When bii is not a diagonalization witness, it never changes. So bi = bsi

for all s ≥ i.

Assume that bii is a diagonalization witness. bii may be changed at most once when

adding dependence relations on bs−1i = bii at some large stage s with As−1 i = As i.

Then we will define a new basis approximation element bsi , and bi = bsi never changes

later since bsi is not a diagonalization witness any more. So the approximations

bsi (s ∈ N) of bi change at most once. Thus, bi = limsbsi exists.

Lemma 5.5 B = bi : i ∈ N ≤T A.

Proof: For any g ∈ G = ∪sGs, the domain of G . If it is not bsi with s ≥ i when

first added into G, then g /∈ B. Otherwise, assume that g = bsi is first added at

stage s.

Case 1. i = s. If i is even, then bii never changes later and g = bi = bii ∈ B. If i

is odd and g = bii is not a diagonalization witness for N -requirements, then g ∈ B.

If g is a diagonalization witness: first, by using A, find the least stage t0 such that

At0 i= A i; second, check whether we add dependence relations on bii during stages

in (i, t0]. If we add dependence relations and redefine approximations for bi, then

g = bi and g /∈ B. If no dependence relations are added on bii, that is, bt0i = bii, then

g = bi ∈ B.

Case 2. i < s. Then i is odd and we change bs−1i at stage s; in this case, g = bsi

is never changed later. So g = bi ∈ B.

By Lemma 5.2, all Pe, Ne hold. Then C is not computable; and for all e, either

≤Ce is not a group order on G or ≤C

e =≤G or ≤Ce =≤∗

G , that is, every C-computable

order on G is computable. So G has no incomputable orders of degree ≤ deg(C).

By Lemma 5.3, C ≤T A. By Lemma 5.5, G has an A-computable basis, so G has

an order of degree ≥ deg(A). Moreover, as G has no orders of degree C, C <T A.


112

Part III

Reverse mathematics, abeliangroups and modules

113

Chapter 6

Reverse mathematics and divisibleabelian groups

In this chapter, we will examine the proof-theoretic strength of several classical

theorems related to countable divisible abelian groups in the program of reverse

mathematics. For more background on abelian groups, refer to [29].

Definition 6.1 (RCA0) Let G be an abelian group. G is divisible if for any g ∈ G

and nonzero n ∈ N, there is a g′ ∈ G such that g = ng′, and we say g is divisible by

n in G.

In this chapter, groups always mean abelian groups.

6.1 Divisible subgroups

Definition 6.2 (RCA0) Let I be an abelian group. I is called injective if for any

monomorphism µ : A → B and any homomorphism φ : A → I, there exists a

homomorphism ψ : B → I such that φ = ψµ.

We now state a theorem related to injective abelian groups in reverse mathe-

matics, as we will use it later to prove Theorem 6.2.

Theorem 6.1 [26] The following are equivalent over RCA0.

(1) ACA0.


114

Chapter 6. Reverse mathematics and divisible abelian groups

Friedman, Simpson and Smith [26] showed that ACA0 proves the statement that

“every divisible subgroup of an abelian group G is a direct summand of G”. We are

interested in the reversal part of this statement. We will complete the work above

of Friedman, Simpson and Smith, by proving the following theorem.

Theorem 6.2 Over RCA0, the following are equivalent:

(1) ACA0.

(2) Every divisible subgroup of an abelian group is a direct summand.

Proof: (1) ⇒ (2) was proved by Friedman, Simpson and Smith in their paper [26]

(Lemma 6.2). For completeness, we provide a proof here.

Let G be an abelian group and D be a divisible subgroup of G. As ACA0

proves that every divisible group is injective, D is injective. Consider the following

diagram with 1D : D → D the identity homomorphism and µ : D → G the identical

embedding:

0 // D

1D

µ // G

ψ

D

Then there is surjective homomorphism ψ with 1D = ψµ, and the kernel of ψ,

ker(ψ), is g ∈ G : ψ(g) = 0, and G = ker(ψ) ⊕ im(ψ) = ker(ψ) ⊕ D. This

completes the proof of (1) ⇒ (2).

(2) ⇒ (1). Fix a one-to-one function f : N → N. Let G be the divisible

abelian group⊕n∈N

Qxn, where elements of G are finite Q-linear sums of elements

from xn : n ∈ N. For m ∈ N, let x′m = x2f(m) + mx2f(m)+1. Then⊕m∈N

Qx′m is a

divisible subgroup of G, denoted by D. D exists in RCA0.

By our assumption (2), D is a direct summand of G. So there is a subgroup K

of G such that G = D ⊕K. For any n, x2n and x2n+1 now can be uniquely written

as

• x2n = y2n + z2n,

• x2n+1 = y2n+1 + z2n+1,

115


where y2n, y2n+1 ∈ D and z2n, z2n+1 ∈ K.

Recall that any element in D can be expressed as a unique Q-linear sum of

elements in x′m : m ∈ N. Let In be the finite set consisting of all indices m where

x′m appears in the expression of y2n or y2n+1. Then

n ∈ range(f) ⇔ ∃m ∈ In[f(m) = n].

We only need to show that if n = f(m0), then m0 ∈ In. As

x′m0= x2n +m0x2n+1 = (y2n +m0y2n+1) + (z2n +m0z2n+1)

and the expressions of elements inD as linear combinations of elements in x′m : m ∈N are unique, we have x′m0

= y2n +m0y2n+1. Then x′m0appears in the expression

of y2n or y2n+1, i.e., m0 ∈ In.

Now range(f) exists, by the ∆01-comprehension.

Classically, every abelian group has the unique largest divisible subgroup which

is the union of all divisible subgroups. This definition is Σ11. We will show that

the existence of the largest divisible subgroup of an abelian group is equivalent to

Π11-CA0 over RCA0. In order to prove the reversal part, we need the following well-

known theorem of Friedman, Simpson and Smith. Recall that an abelian group is

reduced if it has no nontrivial divisible subgroups.

Theorem 6.3 [26] Over RCA0, the following are equivalent.

(1) Π11-CA0.

(2) Every abelian group is a direct sum of a divisible subgroup and a reduced sub-

group.

However, for the special case of torsion-free abelian groups, as shown in Theorem

6.4 below, we show that the existence of the largest divisible subgroup is equivalent

to the arithmetical comprehension.


(1) ACA0.

116


(2) Every torsion-free abelian group has the largest divisible subgroup.

Proof: (1) ⇒ (2). Let G be a countable abelian group. Let

D = g ∈ G : ∀n > 0∃g′ ∈ G[g = ng′].

D exists by arithmetical comprehension.

We will show that D is the largest divisible subgroup of G. Clearly, the zero 0G

of G is in D. For any m, if m divides gn1 and gn2 in G, then m divides gn1 − gn2 in

G. Thus, D is a subgroup of G.

We now show that D is divisible. For any g ∈ D, g is divisible by all nonzero

natural numbers in G. So for any m ≥ 1, there is a hm ∈ G such that g = mhm.

We claim that hm ∈ D. Then g is also divisible by all nonzero natural numbers in

D. Thus, D is a divisible subgroup of G.

Claim. hm ∈ D.

Proof of the claim.

Fix k ≥ 1. As g = mhm = mkhmk, m(hm − khmk) = 0G. Here, the index

mk of hmk is the multiplication of m and k. As m = 0 and G is torsion-free,

hm − khmk = 0G, i.e., hm = khmk. So hm is divisible by k in G. We have showed

that hm is divisible by all nonzero natural numbers in G. By definition, hm ∈ D.

This ends the proof of the claim.

Now we show that D is the largest divisible subgroup of G. Take any divisible

subgroup A of G. For any a ∈ A and n > 0, there is an a′ ∈ A such that a = na′.

Since a′ ∈ G, a ∈ D. Thus A ⊆ D.

(2) ⇒ (1). Let f : N → N be a one-to-one function. Consider the torsion-free

abelian group⊕n∈N

Qxn, denoted by G. Again, elements of G are just finite Q-linear

sums of elements in xn : n ∈ N.

Let H be the set of all elements∑n<m

anxn of G such that for each n, if the

denominator of an has a prime factor pl, then f(l) = n. One can check that H is a

subgroup of G and that H exists in RCA0.

By (2), H has the largest divisible subgroup, say D. Let

D′ = g ∈ H : ∀n > 0∃g′ ∈ H[g = ng′].

117


From the proof of the direction (1) ⇒ (2), we see that D′ is a divisible subgroup of

H and that D ⊆ D′. Then by the choice of D, D′ = D.

We now show n ∈ range(f) ⇔ xn /∈ D.

• If n = f(m), then 1pmxn /∈ H. Note that the only element y in G with xn = pmy

is 1pmxn. Thus no elements y in H satisfy xn = pmy, and xn /∈ D.

• If n /∈ range(f), then 1kxn ∈ H for any k ≥ 1. Thus xn ∈ D.

By ∆01-comprehension, range(f) exists.

Corollary 6.1 Over RCA0, the following are equivalent:

(1) Π11-CA0.

(2) Every abelian group has the largest divisible subgroup.

(3) Every abelian group is a direct sum of a divisible subgroup and a reduced sub-

group.

Proof: By definition, (1) ⇒ (2). By Theorem 6.3 of Friedman, Simpson and

Smith, (3) ⇒ (1). We now show (2) ⇒ (3).

Assume (2). Then every torsion-free abelian group has the largest divisible

subgroup. By Theorem 6.4, this implies ACA0. Now let G be an abelian group.

Again, by (2), it has the largest divisible subgroup, say D. By Theorem 6.2, ACA0

proves G = D ⊕ R for some subgroup R. As D is the largest divisible subgroup, R

is reduced. Hence, (3) holds.

6.2 Two decomposition theorems

Definition 6.3 (RCA0) An abelian group is primary if there is a prime p such that

its elements have orders a power of p.

For each i ∈ N, pi is the i-th prime number, starting with p0 = 2. Let P be the set

of all prime numbers.


118


(1) ACA0.

(2) Every nontrivial torsion group, say T , is a direct sum of primary subgroups.

That is, there is a set I ⊆ P such that T =⊕p∈ITp, where for each p ∈ P ,

Tp = g ∈ T : ∃n[o(g) = pn], and for p ∈ I, Tp = 0T.

Proof: (1) ⇒ (2). We use classical proof in [29]. Let T be a countable nontrivial

torsion group, i.e., T = 0T. Let o(g) be the order of g in T . Because T is torsion,

o(g) is the least nonzero natural number n such that ng = 0T . The function o exists

in RCA0 (o is indeed Σ00).

For every prime p, Tp = g ∈ T : ∃n[o(g) = pn] exists in RCA0. As g ∈ Tp iff

o(g) = pn for some n ≤ o(g). Tp is also Σ00.

We now show that any element of T can be uniquely written as a Z-linear sum

of elements in Tp for various primes p.

Let g ∈ T with order o(g). Factorize o(g) = qr11 · · · qrkk as prime powers. Let

ni =o(g)

qrii

for all i = 1, · · · , k. Then ni, 1 ≤ i ≤ k, are coprime, so there are integers

ai for all 1 ≤ i ≤ k such that a1n1 + · · ·+ aknk = 1. Then,

g = (a1n1 + · · ·+ aknk)g = a1n1g + · · ·+ aknkg.

Because o(nig) =o(g)ni

= qrii , each nig ∈ Tqi . Thus, g ∈∑

1≤i≤kTqi .

We now show that the sum of g is unique. Suppose that there are xi, yi ∈ Tqi

such that g = x1 + · · ·+ xk = y1 + · · ·+ yk. Then xi− yi ∈ Tqi for all 1 ≤ i ≤ k, and

x1 − y1 = (y2 − x2) + · · ·+ (yk − xk).

The order of x1 − y1 is a power of q1, but the order of right hand side is a multipli-

cation of powers of qi with 2 ≤ i ≤ k. Hence, x1 − y1 = 0. Now consider

x2 + · · ·+ xk = y2 + · · ·+ yk.

In a similar way, we can show that x2 = y2, · · · , xi = yi. Thus, xi = yi for all

i = 1, · · · , k. This proves the uniqueness of sums.

Hence, T =⊕p∈P

Tp, and we may have Tp = 0T for some primes p.

119


Now let I = p ∈ P : ∃ g ∈ T − 0T ∃n [o(g) = pn]. Then I is Σ01, so it

exists in ACA0. Moreover, Tp = 0T ⇔ p ∈ I. Hence, I is the desired set such that

T =⊕p∈ITp with Tp = 0T for each p ∈ I.

(2) ⇒ (1). Let f : N → N be a one-to-one function. Let G be an abelian group

generated by xn : n ∈ N with relations pf(m)xm = 0 for each m ∈ N. Then

G =⊕m∈N

Zpf(m)xm, where for any n ∈ N+, Zn = Z/nZ. We view elements of Zn as

the least nonnegative representatives under relation x ∼ y ⇔ x − y ∈ nZ = nk :

k ∈ Z, i.e., for n ≥ 1, Zn = i ∈ N : i ≤ n− 1.Elements of G are of the form

∑i≤naixi for some ai ∈ Zpf(i) with each i ≤ n. So G

exists in RCA0. Clearly, G is torsion.

By assumption (2), there is a set I ⊆ P such that G =⊕p∈IGp, where for each

prime p, Gp = g ∈ G : ∃n[o(g) = pn], and for each p ∈ I, Gp = 0G.By definition of G, each element has order a product of pf(m) for various m. If

n /∈ range(f), Gpn = 0G. If n = f(m) for some m, then pnxm = 0G, so xm ∈ Gpn ,

and thus Gpn = 0G. SoG =

⊕n∈range(f)

Gpn .

Then n ∈ range(f) ⇔ pn ∈ I. By ∆01-comprehension, range(f) exists.

We will consider the decomposition theorem of divisible abelian groups. Before

proving that, we recall a few reverse mathematical results about countable algebra.

Theorem 6.6 [26] Over RCA0, the following are equivalent:

(1) ACA0.

(2) Every abelian group has a torsion subgroup T containing all torsion elements.

(3) Every vector space has a basis.

Definition 6.4 (RCA0)

(1) For 1 ≤ k <∞, Gk denotes the direct sum of k many Gs, that is, Gk =⊕

1≤i≤kG.

(2) G∞ denotes the direct sum of infinitely many Gs, i.e., G∞ =⊕k∈N

G.

120


Proposition 6.1 (RCA0)

(1) For 1 ≤ k < ∞, a direct sum⊕

1≤i≤kGi of abelian groups is divisible iff for all

1 ≤ i ≤ k, Gi is divisible.

(2) A direct sum⊕i∈NGi of abelian groups is divisible iff for all i ∈ N, Gi is divisible.

For a prime number p, the p-quasicyclic group Z(p∞) =

n

pm: n,m ∈ Z

/Z

exists in RCA0.


(1) ACA0.

(2) Every divisible abelian group is isomorphic to a direct sum of the form⊕n∈I


Q

for some I, J ⊆ N, and if I = ∅, then for each n ∈ I, 1 ≤ ln ≤ ∞.

Proof: (1) ⇒ (2). This was already mentioned by Friedman, Simpson and Smith

in paper [26], we now provide a complete proof. Let G be a countable divisible

abelian group. ACA0 proves the existence of torsion subgroup T of G.

We first show that T is a divisible subgroup of G. For g ∈ T and n ≥ 1, as G

is divisible, there is a g′ ∈ G such that g = ng′. Let the order of g be m. Then

mng′ = 0, so the order of g′ is also finite. Thus g′ ∈ T .

By Theorem 6.2, ACA0 proves T is a direct summand of G. Let G = T ⊕ F .

F is a divisible torsion-free subgroup of G. For each x ∈ F and nonzero n ∈ Z,

there is a unique y ∈ F with x = ny and we write y = 1nx. F is a vector space over

the rational field. As ACA0 proves that every vector space has a basis, let X ⊆ F

be a basis of F . Then F =⊕x∈X

Qx, and each Qx ∼= Q as abelian groups.

Now consider the torsion subgroup T . Suppose that T is nontrivial (otherwise,

we have done). By Theorem 6.5, ACA0 proves that T is a direct sum of nontrivial

primary subgroups. That is, there is some Z ⊆ P such that T =⊕p∈Z

Tp, where

121


for each p ∈ Z, Tp = g ∈ T : ∃n[o(g) = pn] = 0G. Since T is divisible, by

Proposition 6.1 in this section, each Tp is divisible.

Fix p ∈ Z, we now decompose Tp.

Let P0 = g ∈ Tp : pg = 0. g ∈ P0 iff g = 0 or o(g) = p, where o(g) is the order

of g in Tp. P0 is Σ00 with parameter Tp, and thus exists in ACA0. Another view is

that P0 = g ∈ Tp : pg = 0 is a Zp-module, where Zp is the finite field of size p.

Similarly, ACA0 proves P0 has a basis Y0 as vector spaces over Zp. So P0 =⊕y∈Y0

Zpy.

List Y0 as y0,1, y0,2, · · · without repetitions. Let Kp = i ∈ N : y0,i ∈ Y0.

For each y0,i ∈ Y0, since Tp is divisible, for each n ≥ 1, there are yn,i ∈ Tp such

that y0,i = pnyn,i. Let Yn := yn,i : i ∈ Kp.

Consider P1 = g ∈ Tp : o(g) = p2. Then pP1 = pg : g ∈ P1 ⊆ P0. We now

show that any element in P1 is a unique sum of elements in Y0 ∪ Y1. Let g ∈ P1,

then pg ∈ P0. There are y0,i1 , · · · , y0,ik ∈ Y0, a1, · · · , ak ∈ Zp such that

pg = a1y0,i1 + · · ·+ aky0,ik

in P0 and the expression of Zp-linear sums is unique.

As y0,il = py1,il for each l = 1, · · · , k and y1,il ∈ Y1, we have

pg = p(a1y1,i1 + · · ·+ aky1,ik).

So g− (a1y1,i1 + · · ·+ aky1,ik) ∈ P0. Then g− (a1y1,i1 + · · ·+ aky1,ik) can be written

as a unique Zp-linear combination of elements in Y0, and thus, g can be written as

a unique Zp-linear combination of elements in Y0 ∪ Y1. So P0 ∪ P1 ⊆⊕

y∈Y0∪Y1Zpy.

In general, let Pn = g ∈ Tp : o(g) = pn+1. Any element in Pn is a unique

Zp-linear sum of elements in∪

0≤j≤nYj. Thus P0 ∪ · · · ∪ Pn ⊆

⊕y∈Y0∪···∪Yn

Zpy. Then

Tp ⊆∪n∈N

Pn ⊆⊕n∈N

⊕y∈Yn

Zpy

=⊕n∈N

⊕i∈Kp

Zpyn,i ⊆ Tp

Now Tp =⊕n∈N

⊕i∈Kp

Zpyn,i ∼=⊕i∈Kp

⊕n∈N

Zpyn,i.

122


For each i ∈ Kp, the subgroup generated by yn,i : n ∈ N is isomorphic to

Z(p∞). Hence, Tp ∼=⊕i∈Kp

⊕n∈N

Zpyn,i ∼=⊕i∈Kp

Z(p∞) = (Z(p∞))|Kp|, where |Kp| is the

size of Kp if Kp is finite, and ∞ otherwise.

This completes the decomposition of Tp for a fixed p ∈ Z.

Finally, we have

G ∼= T ⊕ F ∼=⊕p∈Z

Tp ⊕ F

∼=⊕p∈Z

(Z(p∞))|Kp| ⊕⊕x∈X

Q

Now let I = n ∈ N : pn ∈ Z, J = X, and for each n ∈ I, ln = |Kpn|. Then

G ∼=⊕n∈I


Q.

(2) ⇒ (1). Let f : N → N be a one-to-one function. Consider abelian group⊕m∈N

Z(p∞f(m))xm, denoted by G. Elements of G are of the form∑i≤naixi for some

ai ∈ Z(p∞f(i)) with each i ≤ n. G exists in RCA0, and is divisible.

By (2), G is a direct sum of subgroups isomorphic to Q or Z(p∞) for various

primes p. Let G ∼=⊕n∈I


Q for some I, J ⊆ N, and when I = ∅,

for each n ∈ I, 1 ≤ ln ≤ ∞.

Then I = range(f), J = ∅, and ln = 1 for each n ∈ I. Hence, n ∈ range(f) iff

n ∈ I, so range(f) exists.

123

Chapter 7

Reverse mathematics and modules

7.1 Projective modules

Recall that a R-module P is projective if for every surjective homomorphism ε : B →C of R-modules and every homomorphism g : P → C, there exists a homomorphism

f : P → B such that g = εf .

7.1.1 Direct sums of projective modules

Proposition 7.1 (RCA0) A direct sum⊕

0≤i≤nPi is projective iff Pi is projective for

all 0 ≤ i ≤ n.

However, for the infinite case, we have the following theorem.

Theorem 7.1 (1) (RCA0) If a direct sum P =⊕i∈NPi is projective, then Pi is

projective for all i ∈ N.

(2) (Π11-CA0) If for all i ∈ N, Pi is projective, then the direct sum P =

⊕i∈NPi is

projective.

Proof: We first prove (1) within RCA0. Suppose that P is projective. We will

prove that each Pi is projective. Given any diagram of the form

Pi

gi

Bε // C // 0

124

Chapter 7. Reverse mathematics and modules

Define g : P → C as g(x) = gi(xi) where xi is the i-th column of x ∈ P . As P is

projective, there is a f : P → B such that g = εf . For each z ∈ Pi, we have

εf(⟨0, · · · , 0, z, 0, · · · ⟩) = g(⟨0, · · · , 0, z, 0, · · · ⟩) = gi(z).

So there is a homomorphism fi : Pi → B with fi(z) = f(⟨0, · · · , 0, z, 0, · · · ⟩) such

that εfi = gi.

We now prove (2) within Π11-CA0. Suppose that Pi is projective for all i, and

we will prove that P is also projective. Give a surjective homomorphism ε : B → C

and a homomorphism g : P → C, we need to find a homomorphism f : P → B such

that εf = g.

P

g

f

Bε // C // 0

For each i, let gi : Pi → C be a homomorphism with fi(z) = g(⟨0, · · · , 0, z, 0, · · · ⟩).As Pi is projective, there is a homomorphism fi : Pi → B such that

εfi(z) = gi(z) = g(⟨0, · · · , 0, z, 0, · · · ⟩).

Then for any x = ⟨x0, · · · , xn⟩ ∈ P , set f(x) = f0(x0) + · · ·+ fn(xn), which is in B,

and we have

εf(x) =∑

0≤i≤n

εfi(xi) =∑

0≤i≤n

g(⟨0, · · · , 0, xi, 0, · · · ⟩) = g(x).

So f exists in Π11-CA0. Note that to define f , we use ⟨fi : i ∈ N⟩.

The following theorem has already been studied by Yamazaki in [22]. It provides

several equivalence characterizations of projective modules, and such equivalence are

provable over RCA0. For completeness, we provide a proof here.

Theorem 7.2 (Yamazaki [22]) (RCA0) For a R-module P , the following are equiv-

alent:

(1) P is projective.

(2) If ε : B → P is a surjective homomorphism, then there exists a homomorphism

f : P → B such that εf = 1P , the identity homomorphism on P .

125


(3) If P ∼= B/A, then P is isomorphic to a direct summand of B.

(4) P is isomorphic to a direct summand of a free module F .

Proof: (1) ⇒ (2). As P is projective, there is a homomorphism f : P → B such

that εf = 1P . That is, the diagram

P

1P

f

Bε // P // 0

commutes.

(2) ⇒ (3). Let π : B → B/A be the canonical surjective homomorphism, and

g : B/A→ P be an isomorphism. Then ε = gπ is a surjective homomorphism from

B to P . Let µ : A→ B be the identical embedding, i.e., for any x ∈ A, µ(x) = x.

Consider the short exact sequence 0 → Aµ→ B

ε→ P → 0. By (2), it splits with

a splitting f . Then by Lemma 1.2, B = A ⊕ f(P ). As f is one-to-one, P ∼= f(P ),

which is a direct summand of B.

(3) ⇒ (4). By Proposition 1.7, P ∼= F/A for some free module F . By (3), P is

isomorphic to a direct summand of F .

(4) ⇒ (1). By (4), P is isomorphic to a submodule Q of F via some isomorphism

f , and F = Q ⊕ A for some submodule A. As free modules are projective, F is

projective, and hence, by Proposition 7.1, Q is projective.

Now we show that P is projective. Give a surjective homomorphism ε : B → C

and a homomorphism g : P → C, we need to find a homomorphism h : P → B such

that the diagram

P

g

h

Bε // C // 0

commutes. Consider the homomorphism gf−1 : Q→ C. Since Q is projective, there

is a homomorphism h′ : Q→ B such that gf−1 = εh′. Let h = h′f . Then

εh = ε(h′f) = (εh′)f = (gf−1)f = g(f−1f) = g.

126


7.1.2 Free modules and projective modules over Σ01-PIDs

Free modules are projective, but generally, projective modules may not be free.

Now we restrict to Σ01-PIDs, because over such special rings, projective modules are

always free. Within RCA0, define an integral domain R to be a Σ01-PID if every

Σ01-ideal of R is generated by one element.

Lemma 7.1 (ACA0) Let R be a Σ01-PID. A submodule of a free R-module is free.

Proof: We reason in ACA0. As we only consider countable modules, we assume

that a free R-module F is either⊕

0≤i≤nR for some n ∈ N, or

⊕i∈NR. We prove it for

the case F =⊕i∈NR. The following proof adopts ideas of a classical proof in [30].

Let N be a submodule of F . Fix a code for N . Now elements of N are coded as

natural numbers. For all i, let Ni = N ∩⊕

0≤j≤iR, and define a map fi : Ni → R with

fi(⟨x0, · · · , xi−1, xi⟩) = xi. Then fi is a R-homomorphism, and im(fi) = Ni/Ni−1 is

an ideal of R. Again, we view elements in im(fi) as representatives of equivalence

classes under the equivalence relation x ∼ y ⇔ x − y ∈ Ni−1 such that their codes

are least as natural numbers. im(fi) exists in RCA0.

As R is a Σ01-PID, for each i, there exists a bi such that (bi) = im(fi). Define a

function g by setting g(i) = bi. Since g(i) = bi if bi ∈ im(fi) and

∀a ∈ im(fi)∃c ∈ R[a = cbi],

g is Π02, thus exists in ACA0.

Let J = j ∈ N : g(j) = 0. List J as j0 < · · · jn < jn+1 < · · · . For each jn ∈ J ,

define cjn by primitive recursion as follows. Let cj0 ∈ Nj0 be the element with least

code such that fj0(cj0) = bj0 . Assume that we have defined cjl for all l ≤ k − 1, to

define cjk , we let rjk ∈ Njk be the element with least code such that fjk(rjk) = bjk ,

and check whether the code of rjk is bigger than or equal to the code of cjk−1or not.

If yes, just let cjk = rjk ; otherwise, we can have m big enough such that the code

of mcjk−1+ rjk is bigger than the code of cjk−1

, and we let cjk = mcjk−1+ rjk . This

ensures that the sequence cjk , k ∈ N, has increasing codes, and we still have

fjk(cjk) = fjk(mcjk−1+ rjk) = fjk(rjk) = bjk .

127


C = cj : j ∈ J is ∆01 in J and hence exists in ACA0.

Claim 1. C = cj : j ∈ J is linearly independent.

Proof of claim 1: For any finite subset A ⊆ C, let A = ci1 , ci2 , · · · , cin with

i1 < i2 < · · · < in. If∑

0≤k≤ndikcik = 0 for some dik ∈ R, then

0 = fin(∑

0≤k≤n

dikcik) =∑

0≤k≤n

dikfin(cik) = dinbin .

As bin = 0, din = 0, then∑

0≤k≤n−1

dikcik = 0. By applying fin−1 , we obtain din−1 = 0.

Continue this process, we arrive at din = din−1 = · · · = di1 = 0. So A is linearly

independent.

This completes the proof of claim 1.

Claim 2. C generates N .

Proof of claim 2: We prove it by contradiction. Assume that there is a least

n ∈ N such that some x = ⟨x0, · · · , xn⟩ ∈ Nn that cannot be written as linear

combinations of elements of C.

By the minimality of n, xn = 0. Then fn(x) = xn ∈ (bn) = 0, thus, bn = 0,

and n ∈ J . Let xn = dnbn for some dn ∈ R− 0. Then

fn(x) = xn = dnbn = dnfn(cn) = fn(dncn),

so x− dncn ∈ ker(fn) = Nn−1.

As x cannot be written as linear combinations of elements of C and cn ∈ C, so

does x− dncn. But x− dncn ∈ Nn−1, contradicting the minimality of n.

This completes the proof of claim 2.

By claims 1 and 2, C is a basis of N , and thus N is free.

Theorem 7.3 (ACA0) Let R be a Σ01-PID. A projective R-module is free.

Proof: Since RCA0 proves that every projective module is isomorphic to a direct

summand of a free module, it proves that every projective module is isomorphic

to a submodule of a free module. Let M be a projective R-module. There is an

isomorphism f :M →M ′, whereM ′ is a submodule of a free R-module. By Lemma

7.1, ACA0 proves M′ is free. Let C be a basis of M ′. Then f−1(C) := f−1(c) : c ∈

C is a basis of M , i.e., M is free.

128


Corollary 7.1 (ACA0) Let R be a Σ01-PID. A submodule of a projective R-module

is projective.

7.2 Injective modules

The notion of injective modules is the dual of projective modules. A R-module I is

injective if for any monomorphism µ : A → B and any homomorphism g : A → I,

there exists a homomorphism f : B → I such that g = fµ.

7.2.1 Baer’s criterion for injective modules

The following classical lemma is known as Baer’s criterion for injective modules.

Lemma 7.2 (Baer’s Criterion) A module I over a commutative ring R with identity

is injective iff for every ideal J of R and every R-module homomorphism g : J → I,

there is a homomorphism f : R → I such that f J= g.

By definition, it is easy to see that RCA0 proves one direction: a module I

over a ring R is injective implies that for every ideal J of R and every R-module

homomorphism g : J → I, there is a homomorphism f : R → I such that f J= g.

The other direction is equivalent to ACA0, due to Yamazaki in [22]. For com-

pleteness, we will provide a proof.

Theorem 7.4 (Yamazaki [22]) The following are equivalent over RCA0.

(1) ACA0.

(2) If for every ideal J of R and every R-module homomorphism g : J → I, there

is a homomorphism f : R → I such that f J= g, then I is injective.

Proof: (1) ⇒ (2). We reason in ACA0. Assume that for an ideal J of R, and a

R-module homomorphism g : J → I, there is a homomorphism f : R → I such that

the diagram

0 // J

g

µ // R

f

I

129


commutes, where µ(x) = x for x ∈ J . We need to show that for any submodule A of

a R-module B, the identical embedding µ : A→ B, and a R-module homomorphism

g : A → I, there is a homomorphism f : B → I making the following diagram

commutative.

0 // A

g

µ // B

f

I

List B as b0, b1, · · · , and let An be the R-module generated by A ∪ b0, · · · , bn.We extend g : A→ I to B by stages.

Stage 0. If A0 = A, then let f0 : A0 → I be the same as g. Otherwise, let

S0 = r ∈ R : rb0 ∈ A. Since A is a R-module, S0 is an ideal of R. Define a

homomorphism h0 : S0 → I by letting h0(r) = g(rb0). By assumption, there is a

homomorphism g0 : R → I with g0 S0= h0, g0(1R) ∈ I, and for r ∈ S0,

rg0(1R) = g0(r) = h0(r) = g(rb0).

0 // S0

h0

// R

g0

I

So we define f0 : A0 → I by letting f0(b0) = g0(1R), and for any x = a+rb0 ∈ A0

with a ∈ A, r ∈ R, f0(x) = g(a) + rf0(b0). We claim that f0 is a R-homomorphism

from A0 to I. Let x = a+ rb0 and x′ = a′ + r′b0 be two elements in A0, and r

′′ ∈ R.

Then

f0(x+ x′) = g(a+ a′) + (r + r′)f0(b0)

= g(a) + g(a′) + rf0(b0) + r′f0(b0)

= f0(x) + f0(x′),

and

f0(r′′x) = g(r′′a) + r′′rf0(b0)

= r′′g(a) + r′′rf0(b0)

= r′′f0(x).

130


Moreover, f0 A= g. In fact, if rb0 ∈ A with r = 0, then r ∈ S0, and

f0(rb0) = rf0(b0) = rg0(1R) = g0(r) = h0(r) = g(rb0).

Stage n + 1. Assume that we have fn : An → I extending fn−1. If bn+1 ∈ An,

do nothing, let fn+1 = fn. Otherwise, let Sn+1 = r ∈ R : rbn+1 ∈ An, which is

a proper ideal of R. Let hn+1 : Sn+1 → I by hn+1(r) = fn(rbn+1). Then there is a

R-module homomorphism gn+1 : R → I such that the diagram

0 // Sn+1

hn+1

// R

gn+1

I

commutes. So for any nonzero rbn+1 ∈ An,

fn(rbn+1) = hn+1(r) = gn+1(r) = rgn+1(1R).

Now define fn+1 : An+1 → I by choosing

fn+1(bn+1) = gn+1(1R)

and extending it linearly to all An+1. That is, for x = a + rbn+1 ∈ An+1 with

a ∈ An, r ∈ R, define

fn+1(x) = fn(a) + rfn+1(bn+1).

It is easy to check that fn+1 is the homomorphism from An+1 to I with fn+1 An= fn.

This provides a mapping from B to I by having f(x) = fn(x) for x ∈ An. It is

obvious that f is well-defined and is a homomorphism, and that f is the limit of fn.

(2) ⇒ (1). We prove the reversal direction by showing that (2) implies the

statement “every divisible abelian group is injective” which is equivalent to ACA0

over RCA0.

Let D be a divisible abelian group, (n) be an ideal of Z, and g : (n) → D be

a Z-module homomorphism. We will extend g to Z, i.e., define a homomorphism

f : Z → D such that

f (n)= g.

131


• If n = 0, just define f : Z → D to be the zero homomorphism.

• If n = 0, g(n) ∈ D. By divisibility of D, there is a c ∈ D such that g(n) = nc.

Then define a Z-module homomorphism f : Z → D by letting f(1) = c, and

f(n) = nf(1) = nc = g(n).

So for any x ∈ (n), let x = mxn, then

f(x) = mxf(n) = mxg(n) = g(mxn) = g(x).

Thus, the assumption of (2) is true for R = Z and I = D. Now by applying (2)

to R = Z and I = D, D is injective.

7.2.2 Divisible modules and injective modules over Σ01-PIDs

Let R be an integral domain. Recall that a R-module D is divisible if for every

d ∈ D and every nonzero r ∈ R, there is a c ∈ D such that d = rc. Classically, a

module over a principal ideal domain is injective iff it is divisible. By Proposition

1.11, RCA0 proves that injective modules over an integral domain are divisible.

However, to show the other direction, we require ACA0.

Lemma 7.3 (ACA0) Let R be a Σ01-PID. A divisible R-module D is injective.

Proof: We reason in ACA0. The proof is similar to the one for abelian groups.

Let µ : A → B be a given monomorphism. Without loss of generality, we assume

that A is a submodule of B. For a given homomorphism g : A → D, we need to

find a homomorphism f : B → D such that g = fµ.

0 // A

g

µ // B

f

D

List B as b0, b1, · · · , and let An be the submodule of B generated by A ∪b0, · · · , bn for n ∈ N. An exists by Σ0

1-comprehension. We will extend g to B

by stages.

132


Stage 0. If b0 ∈ A, then A = A0, let g0 = g with domain A0. Otherwise, consider

set

S0 = r ∈ R : rb0 ∈ A.

As A is a R-module, S0 is an ideal of R. Obviously, S0 is ∆01, so by assumption

that R is a Σ01-PID, there is a t0 ∈ R such that S0 = (t0). If t0 = 0, just choose

g0(b0) as 0. If t0 = 0, g(t0b0) ∈ D; by divisibility of D, there is a c0 ∈ D such that

g(t0b0) = t0c0, and define

g0(b0) = c0.

Then for any x ∈ A0, x = a+ rb0 for some a ∈ A and r ∈ R, set

g0(x) = g(a) + rg0(b0).

We can check g0 is a R-module homomorphism from A0 to D, as in Theorem

7.4. We now show that g0 extends g. If rb0 ∈ A with r = 0, then g0(rb0) = rc0. As

r ∈ S0, there is a r′ ∈ R such that r = r′t0. Then

g0(rb0) = rc0 = r′t0c0 = r′g(t0b0) = g(r′t0b0) = g(rb0).

Stage n + 1. Assume that we have already defined gn : An → D extending gn−1.

Now consider bn+1.

If bn+1 ∈ An, then An+1 = An and we let gn+1 = gn. Otherwise, let

Sn+1 = r ∈ R : rbn+1 ∈ An.

Again, Sn+1 is an ideal of R, and Sn+1 is Σ1. By assumption that R is a Σ01-PID,

Sn+1 is a PID and we let Sn+1 = (tn+1). If tn+1 = 0, define gn+1(bn+1) = 0. If

tn+1 = 0, as tn+1bn+1 ∈ An, and hence gn(tn+1bn+1) ∈ D. As D is divisible, there is

a cn+1 ∈ D such that

gn(tn+1bn+1) = tn+1cn+1.

We now define gn+1(bn+1) = cn+1, and extend it linearly to An+1 by letting

gn+1(a+ rbn+1) = gn(a) + rgn+1(bn+1)

for a ∈ An and r ∈ R. We can check that gn+1 is a R-module homomorphism with

domain An+1 and that gn+1 An= gn.

133


Then B =∪n∈N

An, and for all n, gn+1 extends gn. f : B → D defined by

f(x) = gn(x)

for some n with x ∈ An is well-defined and is a R-module homomorphism extending

g. f is the desired homomorphism.

In [26], it was proved that the statement “every divisible abelian group is injec-

tive” is equivalent to ACA0 over RCA0. Together with Lemma 7.3, we obtain:

Theorem 7.5 The following are equivalent over RCA0:

(1) ACA0.

(2) Every divisible module over a Σ01-PID is injective.


Proposition 7.2 (RCA0)

(1) Every quotient of a divisible module over a Σ01-PID is divisible.

(2) Every quotient of an injective module over a Σ01-PID is divisible.

Proof: (1). Let D be a divisible R-module, and ε : D → B a surjective homo-

morphism. We show that B is divisible. For any nonzero b ∈ B, find a d ∈ D with

ε(d) = b. By divisibility of D, for any nonzero r ∈ R, there is a d′ ∈ D such that

d = rd′. Then b = ε(d) = rε(d′) with ε(d′) ∈ B.

(2). Let M be an injective module over a Σ01-PID. RCA0 proves that M is

divisible and that every quotient of a divisible module over a Σ01-PID is divisible.

Hence, RCA0 proves that every quotient of M is divisible.

By Lemma 7.3, ACA0 proves that every divisible module over a Σ01-PID is injec-

tive, and we obtain the following corollary.

Corollary 7.2 (ACA0) Every quotient of an injective module over a Σ01-PID is

injective.

134

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