introduction · 1.1. reverse mathematics. in this paper we only use common subsystems of...

TRANSACTIONS OF THEAMERICAN MATHEMATICAL SOCIETYVolume 360 Number 3 March 2008 Pages 1265ndash1307S 0002-9947(07)04285-7Article electronically published on October 23 2007

RANKED STRUCTURESAND ARITHMETIC TRANSFINITE RECURSION

NOAM GREENBERG AND ANTONIO MONTALBAN

Abstract ATR0 is the natural subsystem of second-order arithmetic in whichone can develop a decent theory of ordinals We investigate classes of struc-tures which are in a sense the ldquowell-founded partrdquo of a larger simpler classfor example superatomic Boolean algebras (within the class of all Boolean al-gebras) The other classes we study are well-founded trees reduced Abelianp-groups and countable compact topological spaces Using computable reduc-tions between these classes we show that Arithmetic Transfinite Recursion isthe natural system for working with them natural statements (such as com-parability of structures in the class) are equivalent to ATR0 The reductionsthemselves are also objects of interest

1 Introduction

Classification of mathematical objects is often achieved by finding invariants fora class of objects ie a method of representing the equivalence classes of some no-tion of sameness (such as isomorphism elementary equivalence bi-embeddability)by simple objects (such as natural numbers or ordinals) A related logical issueis the question of complexity if the invariants exist how complicated must theybe when does complexity of the class make the existence of invariants impossibleand how much information is implied by the statement that invariants of a certaintype exist To mention a far from exhaustive list of examples in descriptive settheory Hjorth and Kechris ([HK95]) investigated the complexity of the existenceof Ulm-type classification (and of the invariants themselves) in terms of the Boreland projective hierarchy see also Camerlo and Gao ([CG01]) and Gao ([Gao04])In computability theory the complexity of index sets of isomorphism relations onstructures has been studied among others by Goncharov and Knight ([GK02])and Calvert ([Cal04]) index sets for elementary equivalence are considered as well(Selivanov [Sel91]) and in reverse mathematics the proof-theoretic strength of thestatement of existence of invariants was studied by Shore ([Sho06])

For some classes closely connected to invariants is the notion of rank For ex-ample the Cantor-Bendixon rank of a countable compact metric space is obtainedby an iterated process of weeding out isolated points This rank then togetherwith the number of points left at the last step constitutes an invariant for thehomeomorphism relation Similar processes of iterating a derivative can be used

Received by the editors August 29 20052000 Mathematics Subject Classification Primary 03F35 03D45 Secondary 03C57 03B30We would like to thank our advisor Richard A Shore for introducing us to questions that are

discussed in this paper and for many useful conversations Both authors were partially supportedby NSF Grant DMS-0100035 This paper is part of the second authorrsquos doctoral thesis

ccopy2007 American Mathematical Society

1265

License or copyright restrictions may apply to redistribution see httpswwwamsorgjournal-terms-of-use

1266 NOAM GREENBERG AND ANTONIO MONTALBAN

to classify well-founded trees superatomic Boolean algebras and reduced Abelianp-groups In this paper we investigate the proof-theoretic strength of various state-ments related to the existence of invariants and ranks on these classes We doit from the viewpoint of Reverse Mathematics we refer the reader to [Sim99] formore information about the program of Reverse Mathematics We assume that thereader is familiar with at least the introductory chapter of [Sim99]

It turns out that in some sense the structures under consideration effectivelycode the ordinals which are their ranks Thus the study of these structures is closelyrelated to two issues the translation processes between these classes which reducestatements about one class to another (and in particular to ordinals) and thestrength of related questions for the class of ordinals The corresponding subsystemof second-order arithmetic is ATR0 the system which allows us to iterate arithmeticcomprehension along ordinals To quote Simpson ([Sim99 76])

ATR0 is the weakest set of axioms which permits the developmentof a decent theory of countable ordinals

Our general aim is to demonstrate that a similar statement can be made for well-founded trees superatomic Boolean algebras etc general statements about theseclasses will be shown to be equivalent to ATR0 thereby implying the necessary use ofordinal ranks in the investigation of these classes Our work continues investigationsof ordinals (Friedman and Hirst [FH90] see [Hir05] for a survey) of reduced Abelianp-groups (see Simpson [Sim99] and Friedman Simpson and Smith [FSS83]) ofcountable compact metric spaces (see Friedman [Fria] and Friedman and Hirst[FH91]) and of well-founded directed graphs (Hirst [Hir00])

As we mentioned key tools for establishing our results are reductions betweenvarious classes of objects These reductions are an interesting object of studyin their own right Indeed we have two points of view classical in which weinvestigate when there are continuous (or even computable) reductions of one classto another and proof-theoretic in which we ask in what system can one show thatthese reductions indeed preserve notions such as isomorphism and embeddability

11 Reverse mathematics In this paper we only use common subsystems ofsecond-order arithmetic The base theory we use will usually be RCA0 ie thesystem that consists of the semi-ring axioms ∆0

1-comprehension and Σ01-induction

We often use the stronger system ACA0 which adds comprehension for arithmeticformulas We note that over RCA0 ACA0 is equivalent to the existence of the rangeof any one-to-one function f N rarr N We will also occasionally use the systemRCA2 which consists of RCA0 together with Σ0

2-inductionThe focus though is Friedmanrsquos even stronger system ATR0 which enables us to

iterate arithmetical comprehension along any well-ordering As mentioned abovethis is the system which is both sufficient and necessary for a theory of ordinals insecond-order arithmetic For example comparability of well-orderings is equivalentto ATR0 over RCA0

For recursion-theoretic intuition we mention that ATR0 is equivalent to thestatement that for every X sub N and every ordinal α X(α) the αth iterate of theTuring jump of X exists

12 The classes We discuss the various classes of objects with which we deal onlybriefly in this introduction as greater detail will be given at the beginning of eachsection The common feature of these classes is that they form the ldquowell-founded


RANKED STRUCTURES AND ARITHMETIC TRANSFINITE RECURSION 1267

partrdquo of a larger class which is simply (arithmetically) definable (whereas the classesthemselves are usually Π1

1) [All structures are naturally coded as subsets of N andso the classes can be considered as sets of reals] The fact that the larger class hasboth well-founded and ill-founded elements will usually imply large complexity theisomorphism relation will be Σ1

1-complete and natural statements about the classwill require Π1

1-comprehension On the other hand when we focus our attentionon the well-founded part the hyperarithmetic hierarchy (and ATR0) suffice

When we discuss each class in detail we specify a notion of isomorphism sim= anda notion of embedding we also define the notion of rank and describe whichstructures are ranked

bull The class of ordinals that is well-orderings of natural numbers (which wedenote by On) is of course a subclass of the class of linear orderings Forembedding we use weak embeddings (one-to-one order-preserving maps)

bull We let WFT denote the class of well-founded trees a subclass of the classof trees of height le ω As the tree structure we take not only the partialordering but also the predecessor relation we thus may assume that alltrees are trees of finite sequences of natural numbers (with the extensionrelation) The notion of embedding only requires preservation of strictorder so an embedding isnrsquot necessarily one-to-one

bull SABA denotes the class of superatomic Boolean algebras (a subclass ofthe class of Boolean algebras) As far as we know this class has not beendiscussed in the setting of reverse mathematics and so we give a detailedtreatment of various definitions and their proof-theoretic content

bull Fixing a prime number p we let R-p-G denote the class of reduced Abelianp-groups a subclass of the class of all p-groups

bull On the analytic side we let CCS denote the class of compact very countabletopological spaces Very countable means Hausdorff countable and secondcountable It turns out that each countable compact Hausdorff space issecond countable but in the setting of second-order arithmetic we canonly treat very countable spaces as reals (so this last statement is notexpressible in this setting) In fact we show that the compact spaces areall metrizable and so the class coincides with countable compact metricspaces However all properties we discuss are purely topological and so weprefer the topological presentation To be strict the class of compact spacesdoes not consist of all ldquowell-foundedrdquo very countable spaces the latterclass (the class of scattered spaces) is larger but ill-behaved so we restrictourselves to the compact case As isomorphisms we take homeomorphismsand as embeddings we take one-to-one continuous and open maps

13 The statements We now discuss the various statements we analyze LetX be a class of structures as above equipped with a notion of isomorphism sim= anotion of embeddability and a subclass of ranked structures

131 Rank For each of the classes we study there is a notion of derivative whichis analogous to the Cantor-Bendixon operation of removing isolated points (such asremoving leaves from trees or eliminating atoms in Boolean algebras by means ofa quotient) Iterating the derivative yields a rank and an invariant which has theexpected properties (for example it characterizes the isomorphism relation and iswell-behaved with regards to the embeddability relation) When dealing with the



class X we define this rank formally and thus the class of structures in X whichare ranked We thus define the following statement

RK(X ) Every structure in X is ranked

We remark that we often show directly that RK(X ) implies other statementsϕ(X ) (without appealing to ATR0)

132 Implications of invariants Suppose that an invariant for isomorphism for theclass X exists Now as this is a third-order statement we follow Shore ([Sho06]) anddiscuss a statement which is immediately implied by this existence Suppose thata sequence AnnisinN of structures in X is given if each An is uniformly assigneda simple object which characterizes its isomorphism type then we could uniformlydecide which pairs (An Am) are isomorphic We thus define

exist-ISO(X ) If 〈An〉nisinNis a sequence of structures in X then the set

(n m) Ansim= Am exists

Suppose that the invariant is even stronger that the simpler objects assigned arequasi-ordered and that the invariant preserves the notion of embeddability Thenas above we could decide the embeddability relation Thus we define

exist-EMB(X ) If 〈An〉nisinNis a sequence of structures in X then the set

(n m) An Am exists

133 Natural statements We define simple statements which are elementary in theanalysis of the class X Relating to Simpsonrsquos words we consider these statements(when true) necessary for the study of X

COMP(X ) For every A and B in X either A B or B A

EQU = ISO(X ) For every A and B in X if AB and BA then Asim=B

134 The structure of the embeddability relation It turns out that for the classesthat we study the embeddability relation is well-founded moreover it forms awell-quasi-ordering whenever 〈An〉nisinN

is a sequence of structures in X there existsome n lt m such that An Am This fact can be added to our list

[Another familiar definition for the notion of well-quasi-orderings is a quasi-ordering which has no infinite descending sequences and no infinite antichainsHowever this equivalence uses Ramseyrsquos theorem for pairs and so cannot be carriedout in our base theory RCA0 In fact the equivalence uses the existence of theembeddability relation (to which Ramseyrsquos theorem is applied) and so by our resultsthe standard proof uses ATR0 (See [CMS04] for a comparison of the differentdefinitions of well-quasi-orderings from the viewpoint of reverse mathematics)]

WQO(X ) The class X quasi-ordered by forms a well-quasi-ordering



14 Reductions As we mentioned reductions between classes of structures pro-vide means of proving equivalences to ATR0 by means of reducing statements fromclass to class However these reductions are interesting in their own right It turnsout that many of the classes in question are as equivalent as they can be

We consider two kinds of reductions One should perhaps be called ldquoeffectiveWadgerdquo reducibility The classes we consider are complicated in the sense thatmembership is often Π1

1-complete Recall that each class X we consider is thewell-founded part of a simpler class Y which is arithmetic ldquoEffective Wadgerdquoreducibility is the analogue of many-one reducibility in the context of sets of reals

Definition 11 For a pair of classes X1X2 which are subclasses of simpler classesY1 and Y2 we say that X1 is EW-reducible to X2 within Y1 and Y2 (and writeX1 leEW X2) if there is some computable functional Φ Y1 rarr Y2 such that Φminus1X2 =X1

The idea is that the question of membership in X1 is effectively reduced to anoracle which gives us membership for X2

Another notion of reducibility is closer to the notion of Borel reducibility forBorel equivalence relations which is extensively investigated by descriptive set-theorists (see for instance [HK01]) Here we consider not the elements of classesX1 and X2 but rather the collection of isomorphism types of these classes and welook for an embedding of one class into the other which is induced by a computabletransformation The structures we work with may have a domain which is a propersubset of ω If we would like to factor out the influence of the complexity of thedomain we arrive at the following definition made by Calvert Cummins Knightand S Miller ([CCKM04]) which is in fact stronger than a mere embedding inducedby a computable function

Definition 12 A computable transformation of a class of structures X1 to anotherclass X2 is a function f X1 rarr X2 for which there is some recursively enumerablecollection Φ such that for all A isin X1 for every finite collection of statements b inthe language of X2 b sub D(f(A)) (the atomic diagram of f(A)) iff there is somefinite collection a sub D(A) such that (a b) isin Φ A computable transformation f isan embedding if f preserves sim= and sim= We write X1 lec X2 if there is a computableembedding of X1 into X2

Another way to think of computable transformations is as functionals which fromany enumeration of D(A) produce uniformly an enumeration of D(f(A))

Computable embeddings (unlike Turing embeddings) preserve the substructurerelation hence preserve embeddability

In some cases we cannot have computable reductions (for example from well-founded trees to ordinals) simply because EQU = ISO(WFT ) fails If instead ofisomorphism classes we consider equimorphism (bi-embeddability) classes we geta slightly different notion of reduction This reduction is not just a one-to-one mapof X1-equimorphism types into X2-equimorphism types it preserves the partialordering on these equivalence classes induced by embeddability

Definition 13 For classes of structures X1 and X2 X1 is equicomputably reducibleto X2 (we write X1 leec X2) if there is a computable transformation f X1 rarr X2

which preserves both and



We introduce notation which indicates that two reductions are induced by thesame function For example X1 leEWc X2 if there is some computable transforma-tion f which is both an EW - and a c-reduction of X1 to X2

15 Results We first consider the proof-theoretic strength of the various state-ments we discussed earlier

Theorem 14RK exist-ISO exist-EMB COMP EQU = ISO WQO

On NA WFT F SABA R-p-G F F CCS

For a statement ϕ and class X a indicates that ϕ(X ) is equivalent to ATR0 overRCA0 A square labelled by ldquoFrdquo indicates that ϕ(X ) is false A square labelled byldquoNArdquo indicates that ϕ(X ) is meaningless

Of course not all of these results are new Friedman and Hirst ([FH90]) showedthat both COMP(On) and EQU = ISO(On) are equivalent to ATR0 over RCA0Shore ([Sho93]) showed that WQO(On) is equivalent to ATR0 over RCA0 Hirst([Hir00]) showed that ATR0 implies RK(WFT ) (actually he proved that every well-founded directed graph is ranked which implies the result for trees) FriedmanSimpson and Smith showed that ATR0 is equivalent to RK(R-p-G) over RCA0 (see[Sim99 Theorem V73]) In [Frib] Friedman shows that over ACA0 WQO(R-p-G)is equivalent to ATR0 and leaves open the question of whether the equivalence canbe proved over RCA0 Shore and Solomon (unpublished) proved that exist-ISO(R-p-G)is equivalent to ATR0 over RCA0 That COMP(CCS) is equivalent to ATR0 (overACA0) can be deduced from results in either [FH91] or [Fria]

Next we turn to reducibilities The classes we deal with are all highly equivalentIn second-order arithmetic we often find that ATR0 shows the existence of reductionsbetween the classes in fact usually what we really use is the fact that structures areranked As this comes for free when the original class is the class of ordinals we canusually show reductions from ordinals to other classes in weaker systems Howeverwe do not have reversals to ATR0 from the statements asserting the existence ofcomputable reductions starting from other classes In particular it is interestingto know if for a class X the existence of a computable reduction from X to theordinals is as strong as ATR0 as we can think of that statement as another way tosay that invariants for X exist

Theorem 15(1) Let X isin WFT SABAR-p-G CCS Then in RCA0 we can show that

On leEWc X In ACA0 we can show that On leEWcec X (2) ATR0 implies the following WFT leEWec On SABA leEWcec On and

R-p-G leEW On

We remark that EW -equivalence of all of our classes follows immediately fromthe fact that On is EW -reducible to every other class for On is Π1

1-completeand these reductions show that each of our classes is Π1

1-complete hence all EW -equivalent The extra information here is that these reductions can be made by



computable transformations (rather than merely Turing reductions) and further-more these transformations often preserve isomorphism nonisomorphism etc

Note that we donrsquot have reductions from CCS to other classes Turing reductionscan be found but computable transformations have not yet been found

We also note that R-p-G leec On This is because is not a total relation onR-p-G

Proofs of the various parts of the theorems appear in the relevant sections

16 More results The last section of this paper is not about Reverse Mathemat-ics as are the previous sections Rather it is about a property shared by all theclasses of structures we study

Clifford Spector proved the following well-known classical theorem in Com-putable Mathematics

Theorem 16 ([Spe55]) Every hyperarithmetic well-ordering is isomorphic to arecursive one

This result was later extended in [Mon05b] as follows

Theorem 17 Every hyperarithmetic linear ordering is equimorphic to a recursiveone

Note that Theorem 17 extends Spectorrsquos theorem because if a linear ordering isequimorphic to an ordinal it is actually isomorphic to it

As for the connection to ATR0 this result can be extended to classes of structuresstudied in this paper For example Ash and Knight mention the following

Theorem 18 ([AK00]) Every hyperarithmetic superatomic Boolean algebra isisomorphic to a recursive one

The following two theorems are straightforward the third less so We give proofsfor all in Section 7

Theorem 19(1) Every hyperarithmetic tree is equimorphic with a recursive one(2) Every hyperarithmetic Boolean algebra is equimorphic with a recursive one

Theorem 110 Every hyperarithmetic compact metric space is isomorphic to acomputable one

Theorem 111 Every hyperarithmetic Abelian p-group is equimorphic with a re-cursive one

We refer the reader to [AK00 Chapter 5] or to [Sac90] for background on hy-perarithmetic theory

2 Ordinals

A survey of the theory of ordinals in reverse mathematics can be found in [Hir05]We follow his notation and definitions As our notion of embedding we take lewan order-preserving injection

We first show below (Proposition 21) that the statement exist-ISO(On) is equivalentto ATR0 over ACA0 we then mention some facts about the Kleene-Brouwer orderingof a tree (these will be useful also in later sections) Using these results we show



(26) that exist-EMB(On) is equivalent to ATR0 over ACA0 Finally (27) we reducethe base to RCA0 for both statements

Equivalences of other statements about ordinals to ATR0 are not new referenceswere made in the introduction

21 Equivalence over ACA0

Proposition 21 (ACA0) exist-ISO(On) is equivalent to ATR0

Proof First assume ATR0 and let αn n isin N be a sequence of ordinals Forall i lt j there is a unique comparison map between αi and αj This shows that(i j) αi equiv αj is ∆1

1-definable By ∆11-comprehension which holds in ATR0

([Sim99 Lemma VIII41]) this set exists Thus exist-ISO(On) holdsSuppose now that exist-ISO(On) holds We will prove COMP(On) which implies

ATR0 Let α and β be ordinals Let

F = (x y) isin α times β α x sim= β y(where α x is the induced ordering from α on the collection of α-predecessorsof x) This set exists by exist-ISO(On) We claim that F itself is a comparison mapbetween α and β

Recall that no ordinal can be isomorphic to any of its proper initial segmentsIt follows that F is a one-to-one function on its domain Furthermore we observethat if (x y) (xprime yprime) isin F then x lt xprime (in α) if and only if y lt yprime (in β) For ifnot say x lt xprime and y gt yprime we compose the isomorphisms α xprime rarr β yprime andβ y rarr α x to get an isomorphism between α xprime and an initial segment of α xAlso not hard to prove is that dom F and range F are initial segments of α and β(this is where we use ACA0)

F is an isomorphism between dom F and range F If they are both proper initialsegments of α and β respectively let α x = dom F β y = range F Then Fwitnesses that (x y) isin F for a contradiction

We could prove that exist-EMB(On) is equivalent to ATR0 over ACA0 using a similarargument Instead we give a different proof

Remark 22 In the following we use effective (∆01)-transfinite recursion in RCA0

The proof that it works is the classical one (using the recursion (fixed-point) the-orem) Also we may perform the recursion along any well-founded relation (notnecessarily linear)

Remark 23 Recall that in RCA0 for every ordinal α one can construct the linearordering ωα In fact for any linear ordering L one can construct ωL in an analogousfashion However one needs ACA0 to show that if α is an ordinal then so is ωαSee Hirst [Hir94]

Recall the following a tree is a downwards closed subset of NltN a tree is well-founded if it does not have an infinite path (all common definitions coincide inRCA0) For linear orderings X and Y T (X Y ) denotes the tree of double descentfor X and Y which consists of the descending sequences in the partial orderingX times Y For a tree T KB(T ) denotes the Kleene-Brouwer ordering on T ([Sim99Section V1]) X lowast Y = KB(T (X Y )) In RCA0 we know that if either X or Yis well-founded then so is T (X Y ) In ACA0 we know that T is well-founded iffKB(T ) is



Lemma 24 (RCA0) Let α be an ordinal and L be a linear ordering Then thereis an embedding of α lowast L into ωα + 1

Proof Let T = T (α L) we know that T is well-founded By effective transfiniterecursion on T we construct for every σ isin T with last element (β l) a recursivefunction iσ Tσ rarr ωβ + 1 (if σ = 〈〉 then β = α) where Tσ = τ isin T σ sube τGiven iσx for every x lt ω (such that σx isin T ) we construct iσ by pasting theseiσxs linearly and placing σ at the end In detail Let S = x isin N σx isin TFor x isin S let βx = (x)0 We have iσ Tσx rarr ωβx For x isin S let

γx =sum

yltNxyisinS

(ωβy + 1)

which is smaller than ωβ for τ isin Tσx let iσ(τ ) = γx + iσx(τ ) Finally letiσ(σ) = ωβ

Now by Π01-transfinite induction on T which holds in RCA0 ([Hir05]) we can

show that for all σ isin T for all τ0 τ1 isin Tσ iσ(τ0) lt iσ(τ1) iff τ0 ltKB τ1

Remark 25 (ACA0) For the next proof we need the fact that if α is an ordinal andL is a non-well-founded linear ordering then α embeds into α lowastL ([Sim99 LemmaV65]) The embedding is obtained by considering T (α) the tree of (single) descentof elements of α We first embed α into KB(T (α)) by taking β lt α to the KB-leastσ isin T (α) whose last element is β Next we embed T (α) into T (α L) by fixing adescending sequence 〈xi〉iisinN

of L and taking 〈β1 βn〉 to 〈(β1 x1) (βn xn)〉To see that this embedding induces an embedding of KB(T (α)) into KB(T (α L))we note that if a ltN b then for all x (a x) ltN (b x)

Proposition 26 (ACA0) exist-EMB(On) is equivalent to ATR0

Proof We show ATR0 by showing the equivalent principle of Σ11-separation ([Sim99

Theorem V51]) Suppose that ϕ ψ are Σ11 formulas which define disjoint classes

of natural numbers From ϕ and ψ we can manufacture sequences 〈Xn〉 and 〈Yn〉of linear orderings such that for all n Xn is a well-ordering iff notϕ(n) and Yn is awell-ordering iff notψ(n) Let

αn = (ωXn + 2) lowast Yn

andβn = Xn lowast (ωYn + 2)

Note that at least one of Xn and Yn is well-founded (and Xn is well-founded impliesωXn well-founded) thus both αn and βn are indeed ordinals

Suppose that Xn is well-founded and that Yn is not We claim that βn embedsinto αn but αn does not embed into βn By Lemma 24 βn ωXn + 1 and byRemark 25 ωXn + 2 αn So βn αn but we cannot have αn βn or we wouldhave ωXn + 2 ωXn + 1 which is impossible

We can thus let A = n βn αn By exist-EMB(On) A exists If ψ(n) holdsthen Xn is an ordinal and Yn is not and so n isin A If ϕ(n) holds then by a similarargument we get βn αn so n isin A as required

22 Proofs of arithmetic comprehension

Proposition 27 (RCA0) Both exist-ISO(On) and exist-EMB(On) imply ACA0



Proof Let ϕ be a Σ01 formula For each n construct an ordinal αn by letting αn

sim= 3if notϕ(n) and αn

sim= 17 if ϕ(n)Now

n ϕ(n) = n αnsim= 17 = n αn 5

exist-ISO(On) implies that the second set exists exist-EMB(On) implies that the thirdset exists

3 Well-founded trees

We denote the class of well-founded trees by WFT If T S are trees then T Sif there is some f T rarr S which preserves strict inclusion Note that f does notneed to preserve noninclusion in fact f may not be injective

The layout of this section is fairly straightforward The standard rank of awell-founded tree is defined in the language of second-order arithmetic we mentionthat Hirst showed that ATR0 implies that every well-founded tree is ranked Wethen show how to get the other (true) statements from RK(WFT ) except forexist-ISO(WFT ) which follows directly from ATR0

We then define the reduction L 13rarr T (L) which maps ordinals to well-foundedtrees and use this reduction to get reversals To get the reduction from trees toordinals we need the notion of a fat tree which we discuss in subsection 322In the last subsection we derive ACA0 from the statements for which the previousreversals required this comprehension

Notation Let T be a tree and σ isin T Then T [σ] = τ isin T τ perp σ T minus σ =τ στ isin T and σT = στ τ isin T

31 Ranked trees

Definition 31 Let T be a tree A node τ isin T is an immediate successor ofa node σ if σ sub τ and |τ | = |σ| + 1 A function rk T rarr α for some or-dinal α is a rank function for T if for every σ isin T rk(σ) = suprk(τ ) + 1 τ is an immediate successor of σ on T and furthermore α = rk(〈〉) + 1 We saythat a tree T is ranked if a rank function of T exists

Lemma 32 (RCA0) Let f T rarr α be a rank function on a tree T Thenrange f = α

Proof Let T be a well-founded tree and rk T rarr α a rank function on it Supposetoward a contradiction that there is a γ lt α not in the range of rk We prove byΠ0

1-transfinite induction that every β such that γ lt β lt α is not in the range of rkThis will contradict that α = rk(〈〉) + 1

Suppose that every βprime between γ and β is not in the range of rk Then for nonode σ can we have rk(σ) = β

Let T be a tree and rk be a rank function on T Let σ isin T By Π01-induction on

|τ | we can show in RCA0 that if σ τ then rk(τ ) lt rk(σ) It follows that

rk(σ) = suprk(τ ) + 1 τ isin T σ τ

Another immediate corollary is

Lemma 33 (RCA0) Every ranked tree is well-founded



(This is true because an infinite path through T would give rise to a descendingsequence in T rsquos rank)

The following two propositions are proved in [Hir00] for well-founded directedgraphs a class which essentially contains the class of trees

Proposition 34 (RCA0) Let T be a tree and let f1 T rarr α1 and f2 T rarr α2 berank functions Then there is a bijection g α1 rarr α2 such that f2 = g f1

Thus ranks are unique up to isomorphism if T is ranked by a function f T rarr αthen we let rk(T ) = α minus 1 = f(〈〉) (Note though that most set theory texts letrk(T ) = α)

Proposition 35 (ATR0) Every well-founded tree is ranked

311 Implications of rank

Lemma 36 (RCA0) Suppose that S and T are ranked trees and that rk(S) rk(T ) Then S T

Proof Let g rk(S) rarr rk(T ) be an embedding of ordinals For each σ isin S wedefine f(σ) isin T by induction on |σ| Along the construction we make sure at everystep that for every σ isin S g(rkS(σ)) le rkT (f(σ)) Let f(〈〉) = 〈〉 Suppose wehave defined f(σ) and we want to define f(τ ) where τ is an immediate successorof σ on S Since g(rkS(τ )) lt g(rkS(σ)) le rkT (f(σ)) there exists π f(σ) withrkT (π) ge g(rkS(τ )) Let f(τ ) be the ltN-least such π

Lemma 37 (ACA0) Let S and T be ranked trees and assume that S T Thenrk(S) rk(T )

Proof Suppose first that there is an embedding f S rarr T we want to constructan embedding g rk(S) rarr rk(T ) Given α lt rk(S) let g(α) = min(rkT (f(σ)) σ isinS amp rkS(σ) = α) We claim that g is an embedding of rk(S) into rk(T ) Considerα0 lt α1 lt rk(S) Let σ isin S be such that rkS(σ) = α1 and g(α1) = rkT (f(σ))Let τ σ be such that rkS(τ ) = α0 Such a τ exists by Lemma 32 applied to SσSince f(τ ) f(σ) g(α1) = rkT (f(σ)) gt rkT (f(τ )) ge g(α0)

Corollary 38 (ACA0) If every well-founded tree is ranked then exist-EMB(WFT )holds

Proof Let 〈Tn〉 be a sequence of well-founded trees Let T =oplus

Tn this is the treeobtained by placing a common root below all of the Tns T = 〈〉 cup

⋃n 〈n〉Tn

The tree T is well-founded and so has a rank function rkT For n m isin N rk(Tn) rk(Tm) iff rkT (〈n〉) le rkT (〈m〉) This is because rkT Tn

is a rank function for Tn (of course we mean 〈n〉Tn) and because for β γ lt rk(T )β γ iff β le γ It follows that Tn Tm iff rkT (〈m〉) le rkT (〈n〉) so the set(n m) Tn Tm exists

Corollary 39 (RCA0) If every well-founded tree is ranked then COMP(WFT )holds

Proof Let T S be well-founded trees let rklowast be a rank function on T oplus S (asbefore this is 〈〉 cup 0T cup 1S) Let α = rklowast(〈0〉) and β = rklowast(〈1〉) Now sinceα β lt rk(T oplus S) either α le β or β le α suppose the former Then rk(T ) rk(S)It follows that T S



Corollary 310 (RCA0) If every well-founded tree is ranked then WQO(WFT )holds


Tn and let rkT

be a rank function on T Now 〈rkT (〈n〉)〉nisinNcannot be strictly decreasing It

follows that for some n le m we have rkT 〈n〉 le rkT 〈m〉 so rk(Tn) rk(Tm) soTn Tm

We have no direct argument to get exist-ISO(WFT ) from RK(WFT ) Rather wegive an argument from ATR0

Proposition 311 (ATR0) exist-ISO(WFT ) holds

Proof We prove that given two recursive trees T and S both of rank α we candecide whether T sim= S recursively uniformly in 0(3α+3) which exists by ATR0 Wedo it by effective transfinite induction For each i let Ti = T minus 〈i〉 Si = S minus 〈i〉Observe that T sim= S if and only if for every i the number of trees Tj such thatTj

sim= Ti is equal to the number of trees Sj such that Sjsim= Ti (this number is

possibly infinite) We can check whether Tisim= Tj and whether Ti

sim= Sj recursivelyuniformly in 0(3α) so we can check whether the sentence above holds recursively in0(3α+3) 312 A reversal We will later get all reversals by translating ordinals into treesHowever we also have one direct reversal akin to the proof for ordinals (Proposition26) it is simpler For any tree T temporarily let 1 + T = 〈〉 cup 〈0〉T As forordinals if T is well-founded then we cannot have 1 + T T for iterating theembedding on 〈〉 would yield a path in T

Proposition 312 (ACA0) exist-EMB(WFT ) implies ATR0

Proof We show Σ11-separation Suppose that ϕ ψ are Σ1

1 formulas which definedisjoint classes From ϕ and ψ we can manufacture sequences 〈Tn〉 and 〈Sn〉 oftrees such that for all n Tn is well-founded iff notϕ(n) and Sn is well-founded iffnotψ(n)

Consider An = Tn times (1 + Sn) and Bn = (1 + Tn) times Sn Both An and Bn arewell-founded for all n Suppose that Tn is well-founded and that Sn is not Wealways have Tn 1 + Tn since Sn is not well-founded we have 1 + Sn Sn (mapeverything onto an infinite path) Thus An Bn

On the other hand again since Sn is not well-founded there is an embeddingof 1 + Tn into Bn (again use an infinite path for the second coordinate) Byomitting the second coordinate we have An Tn It follows that we cannot haveBn An or we would have 1 + Tn Tn We can thus again let the separator ben Bn An 32 Reductions

321 From ordinals to trees

Definition 313 (RCA0) Given a linear ordering L let T (L) be the tree of L-decreasing sequences of elements of L

It is easy to show in RCA0 that for a linear ordering L T (L) is well-foundediff L is so we get an EW-reduction In RCA0 we can show that L 13rarr T (L) is acomputable transformation



Lemma 314 (RCA0) For every α T (α) is ranked and has rank α

Proof For every nonzero σ isin T (α) let rk(σ) be the last element of σ and letrk(〈〉) = α rk is indeed a rank function because for all σ isin T of rank β the set ofranks of immediate successors of σ is exactly all γ lt β

The next corollary follows from Proposition 34 the one after it follows fromLemmas 36 and 37

Corollary 315 (RCA0) For all ordinals α and β α sim= β iff T (α) sim= T (β)

Corollary 316 (ACA0) Let α and β be ordinals Then α β iff T (α) T (β)

Proposition 317 (RCA0) exist-ISO(WFT ) implies ATR0

Proof Corollary 315 shows that exist-ISO(WFT ) implies exist-ISO(On) Proposition 318 (RCA0) RK(WFT ) implies ATR0

Proof We show that exist-ISO(On) holds Let 〈αn〉 be a sequence of ordinals considerT =

oplusT (αn) Let rk be a rank function on T Then αn

sim= αm iff T (αn) sim= T (αm)iff rk(〈n〉) = rk(〈m〉) Proposition 319 (ACA0) exist-EMB(WFT ) COMP(WFT ) and WQO(WFT ) im-ply ATR0

Proof It follows from the previous lemma that each of these three statementsimplies the corresponding one for ordinals and hence ATR0 322 From trees to ordinals We describe fat trees

Definition 320 Given a tree T let Tinfin be the tree consisting of sequences of theform 〈(σ0 n0) (σk nk)〉 where 〈〉 = σ0 σ1 middot middot middot σk isin T and ni isin N

Of course in RCA0 T is well-founded iff Tinfin isLet T be a tree for p isin Tinfin ending with the pair (τ n) we let i(p) = τ We also

let i(〈〉) = 〈〉Lemma 321 (RCA0) Let T be a tree Then (Tinfin)infin sim= Tinfin

Proof We define a map f from Tinfin to (Tinfin)infin by induction At every step we makesure that i(p) = i(i(f(p))) Let f(〈〉) = 〈〉 Suppose we have defined f(q) = σ isin(Tinfin)infin and we want to define f(q〈σ n〉) for some q〈σ n〉 isin Tinfin Let Bqσ =〈σ m〉 isin T times N q〈σ m〉 isin Tinfin Let Aσσ = 〈p m〉 isin Tinfin times N i(p) = σ ampσ〈p m〉 isin (Tinfin)infin Both Bqσ and Aσσ are infinite because σ sup i(q) = i(i(σ)) so we can find a bijection gqσ between them Let f(q〈σ n〉) = σgqσ(〈σ n〉) Definition 322 A tree T is called fat if Tinfin sim= T

Of course a tree is fat iff it is isomorphic to Tinfin for some tree T Note that forall T and p isin Tinfin we have (T i(p))infin sim= Tinfin p It follows that if T is fat andσ isin T then T σ is fat

Also if T is fat then for all σ isin T and for all successors τ of σ in T there areinfinitely many τ prime which are immediate successors of σ and such that T τ sim= T τ prime(Work in Tinfin consider all τ prime such that i(τ prime) = i(τ ))

Lemma 323 (RCA0) Let T be a tree T is ranked iff Tinfin is ranked and in thatcase rk(T ) = rk(Tinfin)



Proof Suppose that T is ranked For all p isin Tinfin let rkinfin(p) = rkT (i(p)) It isimmediate that rkinfin is a rank function on Tinfin and that rk(T ) = rk(Tinfin)

Suppose that Tinfin is ranked by a rank function rkTinfin We note that for allp q isin Tinfin if i(p) = i(q) then rkTinfin(p) = rkTinfin(q) This is because Tinfin p sim= Tinfin qas they are both isomorphic to (T i(p))infin It follows that we can define rkT on Tby letting rkT (σ) = rkTinfin(p) where p is any node such that i(p) = σ It is clearthat rkT is order-inversing If i(p) = σ then the collection of i(q)s of immediatesuccessors q of p is exactly the collection of all extensions of σ in T It follows thatrkT is indeed a rank function on T

Together with Lemmas 36 and 37 we get

Corollary 324 (ACA0) Let S and T be ranked trees Then S T iff Sinfin Tinfin

In fact EQU = ISO holds for fat trees

Lemma 325 (RCA0) Let T be a ranked fat tree Then for every σ isin T andevery γ lt rkT (σ) there are infinitely many immediate successors τ of σ such thatrk(τ ) = γ

(So a ranked fat tree is saturated within its rank)

Proof Work with Tinfin Let p isin Tinfin have rank β we know that rkT (i(p)) = rkTinfin(p)Let γ lt β As the rank function on T i(p) is onto β + 1 there is some extensionτ of σ such that rkT (τ ) = γ Then for every n isin N the immediate extension q ofp which is determined by adding (τ n) as a last pair has rank γ

Corollary 326 (RCA0) Suppose that T and S are ranked fat trees If rk(T ) sim=rk(S) then T sim= S

Proof We define f T rarr S by induction on the levels of T Suppose that f(σ) isdefined For each γ lt rk(σ) consider the set Aσγ which consists of those immediateextensions of σ of rank γ similarly Bσγ consists of those immediate extensions off(σ) of rank γ For all γ lt rk(σ) Aσγ and Bσγ are infinite we can thus buildbijections between them and thus extend f

Next we go from fat trees to ordinals We need the following because T rarr rk(T )is far from computable

Lemma 327 (RCA0) Let T be a ranked fat tree Then KB(T ) is isomorphic toωrk(T ) + 1

Proof We can mimic the construction in the proof of Lemma 24 by effectivetransfinite recursion we construct maps from T σ to ωrk(σ) + 1 which preserveltKB We then use Π0

1-transfinite induction on T to show that each such map isonto its range In fact we can directly compute the final embedding For anyσ isin T let Pσ be the collection of those τ isin T which lie lexicographically to theleft of σ but such that τ |τ | minus 1 sub σ This is in fact finite Order Pσ by ltKB

(which is the same as the lexicographic ordering on Pσ) as 〈τ0 τk〉 We letf(σ) = ωrk(τ0) + 1 + ωrk(τ1) + 1 + middot middot middot + ωrk(τk) + 1 + ωrk(σ)




Proposition 328 (RCA0) exist-EMB(WFT ) implies ACA0

Proof Let f N rarr N be a function We construct a sequence of trees 〈Tn〉nisinN We

have 〈〉 isin Tn for all n furthermore we put 〈x〉 in Tn if f(x) = n Let T = 〈〉Let A = n isin N Tn T We observe that N A = range f Proposition 329 (RCA0) COMP(WFT ) implies ACA0

Proof This is similar to ideas of Friedmanrsquos [Frib] for p-groups Let f N rarr N bea function We construct two trees by defining their leaves Let the leaves of T be〈0〉〈n〉n n isin N let the leaves of S be 〈nn〉〈m〉m n = f(m)cup〈nn〉 n isinrange f

T does not embed in S say g T rarr S is an embedding Let σ = g(〈0〉) Thereis some n such that 〈n〉 sub σ If n isin range f then g(〈022〉) cannot be defined Ifn = f(m) then g(〈0〉〈m + 2〉m+2) cannot be defined

Thus let g S rarr T be an embedding If f(m) = n then g(〈nn〉) must extend〈0k〉 for some k and in fact we must have k gt m because in this case g(〈nn〉〈m〉m)must be defined Thus n isin range f iff for the unique k such that 〈0k〉 sub g(〈nn〉)n isin range f k 331 WQO We will first prove that WQO(WFT ) implies ACA0 using RCA2 andthen show that over RCA0 WQO(WFT ) implies RCA2 (recall that RCA2 is RCA0

together with Σ02-induction) Our proofs of WQO(X ) rarr ACA0 in this and later

sections are motivated by Shorersquos technique of proving WQO(On) rarr ACA0 [Sho93]

Proposition 330 (RCA2) WQO(WFT ) implies ACA0

Proof Let 〈ks〉 be an effective enumeration of 0prime s isin N is a true stage of thisenumeration if for all t gt s kt gt ks s isin N appears to be a true stage at stage t gt sif for all r isin (s t] we have kr gt ks

Let T consist of all sequences σ〈t〉 where t isin N and σ is an increasing enu-meration of the stages which appear to be true at t T is indeed a tree because ift1 lt t2 and t1 appears to be true at t2 then for all s lt t1 s appears to be true att1 iff it appears to be true at t2 We let max σ denote the last element of a sequenceσ isin T

Assume for contradiction that 0prime does not exist Then T is well-founded If f isan infinite path in T then every s isin range f is a true stage since it appears to betrue at unboundedly many later stages

For n isin N let Tn = τ minus τ n τ isin T (ie the elements of Tn are thefinal segments of sequences in T with the first n elements removed) Each Tn is awell-founded tree because T is

For σ isin Tn we say that σ is true if max σ is a true stageBy assumption there exist some n lt m and an embedding g Tn rarr Tm We

claim that the image under g of a true sequence is also true This is because ifτ isin Tm is not true then Tn[τ ] is finite On the other hand for any number r thereis a true string σ isin Tn of length gt r (this requires Σ0

2-induction) and of coursethe true strings on Tn are linearly ordered Thus if σ is true then g(σ) can neverbe off a true string for in that case g would be ldquostuckrdquo

The second point is that if σ isin Tn is true then max g(σ) gt maxσ this is because|g(σ)| ge |σ| and the fact that in Tm we ldquochopped offrdquo more of the beginning of each



string This shows that given any true stage we can manufacture a bigger truestage iterating we compute 0prime

To derive IΣ2 we use ideas of Shore ([Sho93 Theorem 31]) Let ψ(x) =existuforallvφ(x u v) be a Σ0

2 formula and fix n isin N We let Z = x lt n ψ(x)and for p isin N we let Zp = x lt n existu le p forallvφ(x u v) Obviously if p lt qthen Zp sub Zq sub Z Each Zp exists (by bounded Σ0

1-comprehension which holds inRCA0 see [Sim99 Definition II38 and Theorem II39]) so if for some p we haveZp = Z then Z exists This is enough to get induction on ψ up to n

Lemma 331 (RCA0) There is a sequence 〈αp〉pisinNof ordinals such that for all p

if Zp = Z then for all q gt p αq + 1 αp

Proof In Shorersquos construction each αi (Mi for Shore) is defined as a sumsumjltn Nij where Ninminusj is isomorphic to ω3j+1 if notψ(n minus j) and ω3j middot uj minus i +

ω3jminus1 if ψ(n minus j) and uj is the least witness ie the least number u such thatforallvφ(n minus j u v) (If there is a witness a least one exists by Σ0

1-induction) Ofcourse if uj lt i we let uj minus i = 0 Now if Zp Z there exists 0 lt j le n suchthat ψ(n minus j) but notexistu le pforallvφ(n minus j u v) or in other words uj gt p Thensum

k=nminusjnminus1 Nqk + 1 ω3j middot uj minus q + ω3jminus1 middot 2 ω3j middot uj minus p Npnminusj sumk=nminusjnminus1 Npk Then since for all k lt n minus j Nqk Npk we have that

αq + 1 αp

Proposition 332 (RCA0) WQO(WFT ) implies Σ02-induction

Proof Let ψ Z and Zp be as above and let 〈αp〉 be the sequence given by Lemma331

By WQO(WFT ) there exist p lt q such that T (αp) T (αq) In RCA0 wecannot deduce that αp αq but we can deduce that αq + 1 αp This is becauseotherwise we would have 1 + T (αq) T (αq + 1) T (αq) contradicting the well-foundedness of T (αq) Thus Zp = Z and wersquore done

4 Superatomic Boolean algebras

Superatomic Boolean algebras have not as far as we can tell been studied inthe context of reverse mathematics This is why we first discuss various possibledefinitions for this class and see how they relate (for some of the equivalences weseem to require ACA0) We then follow the plan that was executed in the lastsection

41 Definitions

411 Boolean algebras In this subsection unless we mention otherwise we workin RCA0

Definition 41 A Boolean algebra is a set A endowed with two binary operationsand and or a unary operation not and a distinguished element 0A which satisfies thefamiliar axioms of Boolean algebras (For basic properties see [Kop89])

The partial ordering on A given by xory = y (equivalently xandy = x) also existsor and and are indeed the least upper bound and greatest lower bound and so aBoolean algebra is indeed a complemented distributive lattice We use to denotethe symmetric difference operation xy = (xminus y)or (yminusx) where xminus y = xandnoty



The notions of an ideal of a Boolean algebra a homomorphism of Boolean al-gebras products (and infinite sums) of Boolean algebras and subalgebras can becopied from the algebra textbooks and carried out in RCA0 We can also formalizethe notion of the quotient algebra

Definition 42 Let I be an ideal of a Boolean algebra A For a b isin A let a =I b ifab isin I Let B be the collection of a isin A which are the ltN-least elements of their=I -equivalence class B exists as it is ∆0

0 For a b isin B let aorB b = c if aorA b =I cand similarly for andnot 0 Again these operations exist and the resulting structureis a Boolean algebra this is the quotient algebra AI

A particular example is the free Boolean algebra Let V be a set We let Prop(V )be the collection of all propositional formulas with variables in V (this can beeffectively coded) We let hArr denote logical equivalence on propositional formulas(it exists as it is computable by using truth tables) We let B(V ) = Prop(V ) hArrthis is a Boolean algebra which we call the free Boolean algebra over V

In particular we fix some infinite set of variables V lowast and let Prop = Prop(V lowast)For ϕ(x) isin Prop a Boolean algebra A and a isin A ϕA(a) is well-defined (by induc-tion on ϕ) If A is a Boolean algebra and X sub A then we let

〈X〉A = ϕA(a) ϕ isin Prop a isin XIn ACA0 we can show that 〈X〉A (which we call the subalgebra of A generated byX) indeed exists and we can also show that 〈X〉A is the inclusion-wise smallestsubalgebra of A containing X In fact the existence of subalgebras generated bysets is equivalent to ACA0 However we note that if X is finite then in RCA0 wecan show that 〈X〉A exists as there are only 2|X| many propositional formulas withvariables in X

412 Superatomicity in RCA0 We turn to describe superatomic algebras We needsome definitions

Definition 43 Let A be a Boolean algebra An element x isin A is an atom ifx gt 0 but there is no y isin A 0 lt y lt x A Boolean algebra is atomless if it has noatoms

For example if V is infinite then B(V ) is atomlessIf A is a Boolean algebra then we let A+ = A 0A

Definition 44 An embedding of the full binary tree into a Boolean algebra A isa map f 2ltN rarr A+ such that for all σ τ isin 2ltN σ sub τ implies f(τ ) le f(σ) andσ perp τ implies f(σ) and f(τ ) = 0

Lemma 45 (RCA0) If A is atomless then there is an embedding of the full binarytree into A

Proof For σ isin 2ltN we define f(σ) by induction on |σ| Let s(〈〉) = 1A Say thatf(σ) is defined It is not an atom so we can let f(σ0) be the ltN-least y isin A suchthat 0 lt y lt f(σ) and let f(σ1) = f(σ) minus f(σ0)

Definition 46 Let A be a Boolean algebra A set X sub A is free if for all a isin Xand ϕ(x) isin Prop if ϕA(a) = 0 then ϕ is logically false

For example for any set V V is free in B(V )



Lemma 47 (RCA0) Let A be a Boolean algebra Then there is an embedding ofthe full binary tree into A iff there is some infinite free set X sub A

Proof In one direction suppose that X sub A is infinite and free let g N rarr X bea one-to-one enumeration of X For any a isin A let a0 = a and a1 = nota We definef 2ltN rarr A+ by letting f(σ) =

andnlt|σ| g(n)σ(n) That f preserves extension and

incompatibility is immediate the point is that for all σ f(σ) gt 0 This followsfrom freeness f(σ) = 0 because

andnltσ x

σ(n)n is not logically false

In the other direction let f 2ltN rarr A+ be an embedding of the full binarytree into A Let g(n) =

orσisin2n f(σ0) (so g(0) = f(0)) We first show that

g has ldquofree rangerdquo for all distinct n0 nk isin N and for all ϕ(x0 xk) isinProp if ϕA(f(n0) f(nk)) = 0A then ϕ is logically false Let n gt 1 and letϕ(x0 xnminus1) be a propositional sentence For σ isin 2n let ϕσ =

andkltn x

σ(k)k

We have ϕAσ (g(0) g(n minus 1)) = f(σ) For some F sub 2n ϕ is equivalent toor

σisinF ϕσ If ϕ is not logically false then F = 0 take some σ isin F We haveϕA(g(0) g(n minus 1)) ge ϕA

σ (g(0) g(n minus 1)) gt 0Now g is one-to-one it follows that its range contains an infinite set X X must

be free It is clear that if A has an atomless subalgebra then there is an embedding of

the full binary tree into A In fact these notions are equivalent

Lemma 48 (RCA0) Suppose that a Boolean algebra A contains an infinite freeset Then A has an atomless subalgebra

Proof Let X sub A be an infinite free set We construct B sub A as an increasingunion of finite subalgebras 〈Bn〉 which we construct by induction Suppose thatwe have Bn = 〈x0 xnminus1〉A where xi isin X Let mn be the ltN-least naturalnumber which is ltN-greater than all y isin Bn Let xn be the ltN-least element of Xsuch that 〈x0 xn〉A cap 0 mn minus 1 = Bn one exists by freeness if y z isin Xand y z and the xirsquos are distinct then 〈Bn cup y〉A cap 〈Bn cup z〉A = Bn so thepossibilities below mn soon exhaust themselves

Now B =⋃

Bn is isomorphic to the free Boolean algebra on infinitely manyelements and so is atomless

We thus make the following definition

Definition 49 A Boolean algebra is superatomic if it has no atomless subalgebra

Let SABA denote the class of superatomic Boolean algebras

413 Superatomicity in ACA0

Lemma 410 (RCA0) A superatomic Boolean algebra has no atomless quotient

Proof Suppose that a Boolean algebra A has an atomless quotient B = AI LetX sub B be infinite and free Recall that we designed our subalgebras such that assets B sub A So we have X sub A and X is free in A

We work toward the converse Recall that a set X sub A is dense if for all nonzeroa isin A there is some nonzero b isin X such that b le a If I is an ideal of A B is asubalgebra of A and I capB = 0 then the quotient map π A rarr AI is one-to-oneon B and we identify B with its image πldquoB (which exists under ACA0)



Lemma 411 (ACA0) If B is a subalgebra of a Boolean algebra A then there isan ideal I of A such that I cap B = 0 and B is dense in AI

Proof Let B be a subalgebra of A Let 〈an〉nisinNbe an enumeration of the elements

of A We define a sequence of ideals In of A such that for all n In cap B = 0We let I0 = 0A At stage n given In we ask if an bounds some element of B+

modulo In If so we let In+1 = In otherwise we let In+1 be the ideal generatedby In together with an

Now the sequence 〈In〉nisinNexists because each In is finitely generated and we

can keep track of the finite sets of generators Thus we can let I =⋃

n In By theconstruction I cap B = 0 Also B is dense in AI because each element at itsturn is either discovered to bound some element of B+ modulo an ideal containedin I or is thrown into I Corollary 412 (ACA0) A Boolean algebra is superatomic iff it has no atomlessquotient

Proof Suppose that A is not superatomic it has some atomless subalgebra B LetI be given by Lemma 411 Then AI is atomless because B is a dense atomlesssubalgebra of AI Question 413 Does the statement of Corollary 412 imply ACA0

Remark 414 Lemma 411 is equivalent to ACA0 over RCA0

42 Ranked Boolean algebras In this section we define rank functions on su-peratomic Boolean algebras this follows Simpson [Sim99 Section V7] where re-duced abelian p-groups and their Ulm resolutions are discussed

Let B be a Boolean algebra The Cantor-Bendixon derivative of B is the quotientBI where I is the ideal generated by the atoms of B I is called the Cantor-Bendixon ideal of B

Let α be an ordinal A partial resolution of B along α is a sequence of ideals〈Iβ〉βltα such that I0 = 0B if β + 1 lt α then Iβ+1 is the (pull-back to B) ofthe CB-ideal of BIβ and for limit β lt α Iβ =

⋃γltβ Iγ A resolution of B is a

partial resolution of B along some ordinal α+1 such that Iα = B and for all β lt αIβ = B

Suppose that 〈Iβ〉βleα is a resolution of a Boolean algebra B We define as-sociated rank and degree functions For x isin B rk(x) is the unique β lt α suchthat x isin Iβ+1 Iβ and deg(x) = n if in BIrk(x) x is the join of n many atomsWe let inv(x) = (rk(x) deg(x)) we let rk(B) = rk(1B) deg(B) = deg(1B) andinv(B) = inv(1B)

Definition 415 A Boolean algebra is ranked if it has some resolution such thatthe associated invariant function exists

Lemma 416 (RCA0) Let B be a superatomic Boolean algebra Let α αprime be or-dinals and suppose that 〈Iβ〉βleα and 〈I primeβ〉βleαprime are two resolutions of B Furtherassume that the associated invariant functions inv and invprime exist Then α sim= αprimeand the isomorphism commutes with rk rkprime (ie if f α rarr αprime is the isomorphismthen f rk = rkprime)

Proof We remark that for all β lt α βprime lt αprime if Iβ sub I primeβprime then Iβ+1 sub I primeβprime+1 Fortake some atom x of BIβ For all z le x either z isin Iβ or z =Iβ

x It follows that



for all z le x either z isin I primeβprime or z =Iprimeβprime x It follows that x isin I primeβprime or x is an atom of

BI primeβprime Thus every finite join of atoms of BIβ is in I primeβprime or is a finite join of atomsof BI primeβprime

Of course rk and rkprime are symmetric here Thus if Iβ = I primeβprime then Iβ+1 = I primeβprime+1Using Π0

1-transfinite induction on β lt α we show that for all x isin B if rk(x) = βthen Iβ = I primerkprime(x)

Suppose that the claim is verified up to β let x isin B be such that rk(x) = βLet βprime = rkprime(x) We need to see that Iβ = I primeβprime

Let y isin Iβ let γ = rk(y) Now γ lt β so x isin Iγ+1 Let γprime = rkprime(y) By inductionIγ = I primeγprime and so Iγ+1 = I primeγprime+1 and so x isin I primeγprime+1 as x isin Iβprime+1 it follows that γprime lt βprimeThus Iβ sub I primeβprime

Next we note that for no γprime lt βprime can we have Iβ sub I primeγprime for then we would haveIβ+1 sub I primeγprime+1 sub I primeβprime contrary to rkprime(x) = βprime

Let γprime lt βprime Then Iβ sub I primeγprime so there is some y isin Iβ such that rkprime(y) ge γprimeLet δ = rk(y) δ lt β so by induction Iδ+1 = Irkprime(y)+1 Thus Iγprime+1 sub Iβ AsI primeβprime =

⋃γprimeltβprime I primeγprime+1 we have I primeβprime sub Iβ as required

Now we can define a function f α rarr αprime by letting f(β) = βprime if for some (all) xsuch that rk(x) = β we have rkprime(x) = βprime The function f exists as it is ∆0

1-definableWe show that f is an isomorphism Of course dom f = α as the sequence of ideals〈Iβ〉βleα is strictly increasing By the same argument range f = αprime f is order-preserving say γ lt β lt α Then Iβ = I primef(β) and Iγ = I primef(γ) Also Iγ Iβ Itfollows that I primef(γ) I primef(β) and so f(γ) lt f(β)

Corollary 417 (RCA0) Suppose that A B are ranked Boolean algebras and thatf A rarr B is an isomorphism Let rkA A rarr α and rkB B rarr αprime be the rankfunctions Then α sim= αprime and the isomorphism g commutes with rkA rkB f (ierkB f = g rkA)

Lemma 418 (RCA0) A ranked Boolean algebra is superatomic

Proof Let B be a ranked Boolean algebra of rank α Suppose toward a contradic-tion that B is not superatomic Then there is an embedding f of the full binarytree into B By recursion we construct a decreasing sequence 〈σn〉nisinN

sub 2ltω suchthat 〈inv(f(σn))〉nisinN

is a descending sequence in αtimesω getting a contradiction Letσ0 = 〈〉 Given σn either inv(f(σ

n 0)) lt inv(f(σn)) or inv(f(σn 1)) lt inv(f(σn))

for if inv(f(σn)) = inv(f(σn 0)) = inv(f(σ

n 1)) = (β n) then necessarily f(σn) =f(σ

n 0) = f(σn 1) in BIβ contradicting f(σ

n 0) and f(σn 1) = 0B Let σn+1 be one

of σn 0 or σ

n 1 whichever has smaller invariant

Lemma 419 (ATR0) Every superatomic Boolean algebra is ranked

Proof Let B be a Boolean algebra and assume that there is no ordinal α such thata full iteration of the derivative of B along α + 1 exists

Let ϕ(L 〈Ia〉aisinL) denote that L is a linear ordering that for all a isin L Ia is aproper ideal of B and that if a ltL b then the (pull-back to B of the) CB-ideal ofBIa is contained in Ib

For every ordinal α by arithmetic transfinite recursion we can construct aniteration 〈Iβ〉βltα of the derivative of B along α Then ϕ(α 〈Iβ〉βltα) holds



Since the collection of ordinals is not Σ11-definable (this is provable in ACA0 see

[Sim99 V19]) there is some linear ordering L which is not well-founded and suchthat there is a sequence 〈Ia〉aisinL such that ϕ(L 〈Ia〉aisinL) holds

Let a0 gtL a1 gtL a2 gtL be an infinite descending sequence in L LetI =

⋂nisinN

Ian Then BI is atomless (showing that B is not superatomic) For if

y isin B I then for some n y isin Ian If in BI y is an atom then y is an atom in

BIan+1 in which case we would have y isin Ian

Thus if B is superatomic then there is a full iteration 〈Iβ〉βleα along some ordinalα + 1 For every β lt α the collection of atoms of BIα exists Given any b isin Bwe can find the unique β lt α such that b isin Iβ+1 Iβ then find the finite set F ofatoms of BIβ such that b = orF Then the invariant of b in B is (β |F |)

421 Implications of rank The following is a converse to Corollary 417

Lemma 420 (RCA0) Suppose that A B are ranked Boolean algebras which havethe same CB invariant Then A sim= B

Proof This is a back-and-forth construction we define f A rarr B Let f(0A) = 0B

and f(1A) = 1BLet a0 isin A We look for b0 isin B such that invA(a0) = invB(b0) and invA(nota0) =

invB(notb0) Why does such a b0 exist We first note that invA(a0) lelex inv(A) andthat for all pairs (γ n) lelex inv(B) there is some c isin B such that invB(c) = (γ n)Next we note that for all (γ n) ltlex inv(A) there is a unique (β m) lelex inv(A)such that for all a isin A such that invA(a) = (γ n) we have invA(nota) = (β m)By replacing a0 by its complement if necessary we can assume that invA(a0) ltlex

inv(A) Thus we can pick any b0 isin B such that invB(b0) = invA(a0) We letf(a0) = b0 and f(nota0) = notb0

We now repeat the process backward in the other direction in the Booleanalgebras B(le b0) and B(le notb0) we pick new elements find their equivalents inA(le a0) and A(le nota0) as above and extend f to be defined on the subalgebra ofA generated by all the elements picked so far We then repeat inside the four newsmaller algebras we got and so forth

Lemma 421 (RCA0) A (finite or infinite) direct sum of superatomic Booleanalgebras is superatomic

Proof Let A and B be Boolean algebras and suppose that f 2ltN rarr A times B isan embedding of the full binary tree into A times B write f(σ) = (aσ bσ) There aretwo possibilities Suppose that there is some σ such that aσ = 0A Then we candefine g 2ltN rarr B by letting g(τ ) = bστ For all τ g(τ ) gt 0 as f(στ ) gt 0 Ifτ1 perp τ2 then στ1 perp στ2 so f(στ0) perp f(στ1) so g(τ0) perp g(τ1) Thus B is notsuperatomic Similarly if for some σ we have bσ = 0 then A is not superatomic

Otherwise we can let g(σ) = aσ By assumption g(σ) gt 0 for all σ and ifτ0 perp τ1 then f(τ0) perp f(τ1) which implies g(τ0) perp g(τ1) Then A (and B) are notsuperatomic

Let 〈Bn〉nisinNbe a sequence of Boolean algebras and let B =

oplusnisinN

Bn Recallthat the elements of B are those sequences of

prodn Bn which have either almost all

elements 1 or almost all elements 0 Suppose that f 2ltN rarr B is an embedding ofthe full binary tree into B Then for either f(0) or f(1) almost all elements are0 which essentially means that either f(0) or f(1) (and all extensions) lie in some



finite product Thus some finite product of the Bns is not superatomic so someBn is not superatomic Corollary 422 (RCA0) Suppose that every superatomic Boolean algebra is rankedThen exist-ISO(SABA) holds

Proof Let 〈Bn〉 be a sequence of superatomic Boolean algebras Let B =oplus

n BnThen B is superatomic let invB be the invariant function for B We claim thatinvB Bn is the invariant function for Bn (where we identify b isin Bn with thesequence containing b and otherwise only 0s) by Π0

1-transfinite induction on β ltrk(B) we can see that Iβ(Bn) = Iβ(B) cap Bn For the successor step note that Bn

is an initial segment of B at limit stages take unionsBy Corollary 417 and Lemma 420 Bn

sim= Bm iff invB(Bn) = invB(Bm) Thisis equality of elements of inv(B) rather than merely isomorphism of ordinals weuse the fact that if β γ lt α then β sim= γ iff β = γ Thus (n m) Bn

sim= Bmexists

If we care for only one direction we have the following

Lemma 423 (RCA0) Suppose that A B are ranked Boolean algebras and thatinv(A) lelex inv(B) Then there is an embedding of A into B

The proof is similar to the proof of Lemma 420 without going back f insteadof preserving the invariant simply does not decrease it (lexicographically)

Corollary 424 (RCA0) Assume that every superatomic Boolean algebra is rankedThen COMP(SABA) holds

Proof Let A B be superatomic Boolean algebras get an invariant inv on A times BWe know that either inv(1A) lelex inv(1B) or vice versa thus A B or B A Corollary 425 (RCA0) Assume that every superatomic Boolean algebra is rankedThen WQO(SABA) holds


n Bn

and let inv be an invariant on B Since 〈inv(1Bn)〉 cannot be a strictly ltlex-

decreasing sequence we must have some n lt m such that inv(1Bn) lelex inv(1Bm

)It follows that Bn Bm

The analog of Corollary 417 (ie the converse of Lemma 423) seems to requireACA0

Lemma 426 (ACA0) Let A B be ranked Boolean algebras such that A embedsinto B Then rk(A) rk(B)

Proof Let 〈Iβ〉βleα be the CB resolution for A and let 〈Jβ〉βleαprime be the CB resolutionfor B Let f A rarr B be an embedding Define g(β) = βprime if βprime is the least ordinalbelow αprime such that there is some x isin A such that rk(x) = β and rk(f(x)) = βprime

Now we prove that g is order-preserving Let β lt γ lt α Take x isin A suchthat rk(x) = γ and g(γ) = rk(f(x)) x isin Iβ+1 so in AIβ there are infinitely manyatoms below x By induction we can pick an infinite collection X sub A such thatfor all y isin X y le x and in AIβ y is an atom (so rk(y) = β) furthermore fordistinct y yprime isin X we have y and yprime = 0 For all y isin X f(y) isin Jg(β) f(y) le f(x) andthe f(y)s are pairwise disjoint Also f is one-to-one so fldquoX is infinite It followsthat f(x) isin Jg(β)+1 and hence that g(β) lt g(γ)



Remark 427 (ACA0) Suppose that A B are ranked Boolean algebras and thatrk(A) = rk(B) Then A B iff deg(A) le deg(B) For if deg(A) le deg(B)then inv(A) lelex inv(B) so A B (Lemma 423) Suppose that f A rarr B isan embedding let α = rk(A) and n = deg(A) Let X sub A be a set of size n ofpairwise disjoint elements of rank α By Lemma 426 for each x isin X rkB(f(x)) =rk(B(le f(x))) ge rk(A(le x)) = rkA(x) = α so in B there are n pairwise disjointelements of rank α it follows that deg(B) ge n

Corollary 428 (ACA0) Assume that every superatomic Boolean algebra is rankedThen exist-EMB(SABA) holds

Proof The proof is similar to that of Corollary 422 We are given a sequence〈Bn〉 of superatomic Boolean algebras and get a rank on B =

oplusn Bn Now if

inv(Bn) lelex inv(Bm) then Bn embeds into Bm On the other hand if Bn embedsinto Bm then rk(Bn) rk(Bm) and if they are equal then deg(Bn) le deg(Bm)However as all of these ordinals are initial segments of rk(B) and le coincideso if Bn embeds into Bm then inv(Bn) lelex inv(Bm) The conclusion follows Corollary 429 (ACA0) Assume that every superatomic Boolean algebra is rankedThen EQU = ISO(SABA) holds

Proof Let A B be superatomic Boolean algebras such that A B and B AAgain let inv be an invariant on AtimesB By Lemma 426 and Remark 427 we havethat inv(1A) = inv(1B) It follows that A sim= B 43 Reductions

431 Ordinals to superatomic Boolean algebras Let L be a linear ordering Welet Int(L) be the Boolean algebra consisting of finite unions of half-open intervalsof L of the form [a b) (where we allow b = infin) [In RCA0 the elements of thisBoolean algebra are coded by the finite sequences of the pairs of endpoints of theseintervals all Boolean operations exist]

For any linear ordering ordinal L let B(L) = Int(ωL)

Lemma 430 (RCA0) For all α B(α) is ranked and its invariant is (α 1)

Proof The important thing to recall is that the operations of ordinal addition andsubtraction (αminusβ = γ if β+γ = α) below ωα exist also taking the logarithm of baseω exists That is given γ = ωβ1n1 +ωβ2n2 + middot middot middot+ωβknk where β1 gt β2 gt middot middot middot gt βk

are elements of α and ni isin ω we know that β1 is the greatest element β of αsuch that ωβ le γ we in fact let inv(γ) = (β1 n1) and for every interval [β γ)(where β lt γ le ωα) we let inv([β γ)) = inv(γ minus β) Note that for the special caseγ = infin (= ωα) we always have ωα minus β = ωα and inv(ωα) = (α 1) When faced bya finite disjoint union of intervals we take the natural ldquosumrdquo of invariants namelythe invariant is (β n) where β is the maximal rank of the intervals and n is thesum of the degrees of those intervals which have rank β

The fact that inv is indeed the correct invariant for B(α) follows from three facts(all of which are properties of ordinal addition up to ωα) one that finite unionsdo not increase rank second that the inclusion relation respects the lexicographicordering of invariants (so in particular Iγ the collection of all elements of B(α)which have rank below γ forms an ideal) and third that an interval of length ωγ

cannot include two disjoint subintervals of length ωγ this implies that the intervalsof invariant (γ 1) (which modulo Iγ have length ωγ) are the atoms of B(α)Iγ



Lemma 431 (RCA0) For all linear orderings L L is well-founded iff B(L) issuperatomic

Proof The direction from left to right follows from the previous lemma and fromLemma 418 For the other direction suppose that L is not well-founded and thatannisinN is a descending sequence in L We define an embedding f of the full binarytree into B(L) Given σ isin 2ltN we let xσ =

sumiσ(i)=1 ωai+1 and we let

f(σ) = [xσ xσ + ωa|σ|)

Is not hard to check that f is as wanted We get (aided by Corollary 417 and Lemma 420)

Corollary 432 (RCA0) Let α β be ordinals Then α sim= β iff B(α) sim= B(β)

The following is immediate

Lemma 433 (RCA0) Let α β be ordinals and suppose that α β Then B(α) B(β)

By Lemma 426 we get

Corollary 434 (ACA0) Let α β be ordinals Then α β iff B(α) B(β)

As a corollary we have

Corollary 435 (RCA0) The statement exist-ISO(SABA) implies exist-ISO(On) andEQU = ISO(SABA) implies EQU = ISO(On) Therefore both exist-ISO(SABA) andEQU = ISO(SABA) are equivalent to ATR0

Corollary 436 (ACA0) The statement exist-EMB(SABA) implies exist-EMB(On) thestatement COMP(SABA) implies COMP(On) and the statement WQO(SABA) im-plies WQO(On) Therefore exist-EMB(SABA) COMP(SABA) and WQO(SABA)are equivalent to ATR0 over ACA0

432 Well-founded trees to superatomic Boolean algebras We do not need the fol-lowing for the proof of Theorem 15 but we thought to include a natural operationwhich takes us directly from trees to Boolean algebras

Let T be a tree We let B(T ) be the tree algebra of T as described in [Kop89Section 16] There it is defined as the subalgebra of P(T ) generated by the conesσ isin T τ sube σ (for all τ isin T ) Of course this definition only makes sensein ACA0 but in fact we can get the tree algebra in RCA0 We let B0(T ) be aBoolean algebra generated freely over a basis bσ σ isin T We let I be the idealgenerated by the equations bσ le bτ if τ sub σ and bσ and bτ = 0 if σ perp τ The idealI exists by reducing to disjunctive normal form it is sufficient to decide whetheran element of the form a = bσ1 and middot middot middot and bσn

andminusbτ1 and middot middot middot and minusbτmis in I If the σi are

not linearly ordered then a isin I Otherwise we let σ be the greatest σi moduloI a = bσ and minusbτ1 and middot middot middot and minusbτm

If for some i we have σ sup τi then a isin I otherwisea isin I Finally we let B(T ) = B0(T )I

Remark 437 The argument showing that I exists and some more work yield anormal form for elements of B(T ) all the elements can be written as disjoint sumsof elements of the form bσ minus

sumτisinS bτ where S is a finite antichain in T and for all

τ isin S σ sub τ Again see [Kop89]

Recall that for a tree T and τ isin T T minus τ = σ τσ isin T



Fact Let T be a tree For all τ isin T B(T minus τ ) sim= B(T )(le bτ )

In fact we can break up the tree algebra into pieces Let τ isin T and let S(τ ) bethe collection of immediate successors of τ on T If S(τ ) is infinite then

B(T minus τ ) sim=oplus

B(T minus σ) σ isin S(τ )

However if S(τ ) is finite (for example if τ is a leaf) then bτ minussum

σisinS(τ) is an atomof B(T minus τ ) and so we have

B(T minus τ ) sim= 0 1 oplus B(T minus σ) σ isin S(τ )(So if τ is a leaf then bτ is an atom of B(T ) and B(T minus τ ) sim= 0 1)

If T is well-founded then we can build B(T ) from the leaves up

Lemma 438 (ACA0) If T is well-founded then B(T ) is superatomic

Proof Suppose that f is an embedding of the full binary tree into B(T ) Byinduction we find some σn isin T and some embedding fn of the full binary treeinto B(T σn) Given fn(〈〉) = bσn

= 1B(Tminusσn) for some i lt 2 there is somefinite S sub S(σn) such that fn(i) le

sumτisinS bτ We can thus view ρ 13rarr fn(iρ) as an

embedding of the full binary tree intooplus

B(T minus τ ) τ isin S By asking finitelymany questions as in the proof of Lemma 421 we find some coordinate from whichwe can pick a σn+1 and let fn+1 be the adequate restriction

It is clear that if T sim= T prime then B(T ) sim= B(T prime)

433 Superatomic Boolean algebras to well-founded trees

Definition 439 A uniformly splitting tree is a tree of the form σ isin ωltα foralli lt|σ|(σ(i) lt mi) where α isin Ncup N and for all i mi isin N and mi ge 2 A finite treeembedding into a Boolean algebra B is a partial function g T rarr B+ where T is afinite uniformly splitting tree and g preserves le perp and 1 (ie g(〈〉) = 1B)

Fix a sequence of constants 〈qn〉nge2 which grows sufficiently quickly so that for

all l k lt n ql + qk le qn (for example let qk = 22k

) Also fix an infinite recursiveset C = cn n ge 2 (say cn = 〈0 n〉) and fix a coding of all finite sequences andfunctions such that the collection of code numbers is disjoint from C For m isin Nn ge 2 we let 13(n m) = 〈cn〉m and let 13n = 13(n qn minus 1)

Given a Boolean algebra B we will code all finite tree embeddings into B usinga tree Let T sub m1 timesmiddot middot middottimesmk be a finite uniformly splitting tree and let g T rarr Bbe a finite tree embedding For l le k let Tl = T cap ωl We let the code for g be thestring

(g) = 13m1〈g T1〉13m2

〈g T2〉 middot middot middot 13mk

〈g Tk〉Note that we also allow g to be defined on T0 = 〈〉 there is a unique such g

(we must have g(〈〉) = 1B) and in this case (g) = 〈〉 The function g 13rarr (g) isone-to-one and its range is computable from B so it exists (in RCA0)

The tree T (B) consists of all of the strings (g)13(n m) where n ge 2 andm lt qn and g is a finite tree embedding into B In RCA0 we can show that T (B)exists Also T (B) is closed under initial segments so it is indeed a tree

Lemma 440 (RCA0) B is superatomic iff T (B) is well-founded



Proof Suppose that B is not superatomic Then there is an embedding of the fullbinary tree into B and by shifting elements around we may assume that the topof the embedding is 1B This embedding would yield a path in T (B)

On the other hand from a path in T (B) we can recover an embedding of aninfinite uniformly splitting tree into B The reason is that on an infinite pathcodes for functions must occur infinitely often if f is an infinite path and f(n) is acode for a function then if cm = f(n + 1) we know that f(n + qm) must be a codefor a function Restricting the embedding we get to 2ltω we get an embedding ofthe full binary tree into B

For the purpose of the following computation we let q1 = 0 Also recall thatI0 = 0 so that rk(0B) = minus1 but for every finite join of atoms x rk(x) = 0

Lemma 441 (RCA0) Suppose that B is a ranked Boolean algebra and inv(B) =(α n) Then T (B) is ranked and has rank ω(2α + 1) + qn

(Recall that if γ is a limit ordinal and α = γ + k then 2α = γ + 2k)

Proof For a finite tree embedding g into B we let inv(g) = mininv(x) x isinrange g where the ordering is of course the lexicographic one For each such g ifinv(g) = (β k) then we let f((g)) = ω(2β + 1) + qk

Now we claim that f extends to a rank function on T (B) We first discusshow we should define f on nodes of the form (g)

13n for n ge 2 and finite treeembeddings g into B Let g be such and let (β k) = inv(g) let x isin range g suchthat inv(x) = (β k) Let n ge 2 If n gt k then we cannot split x into n manydisjoint elements of rank β on the other hand we can split x into n many disjointelements each of which have invariant below (β 0) but as large as we like below(β 0) It follows that the immediate extensions of (g)13n in T (B) are of theform (gprime) where inv(gprime) lt (β 0) (so if β = 0 there are no such gprime and (g)13n

is terminal) and all invariants below (β 0) occur then the supremum of f on theimmediate extensions of (g)

13n is supω(2γ + 1) + l γ lt β l isin N = ω(2β)We thus let f((g)

13n) = ω(2β) for such nIf however n le k then x can be split into n disjoint elements of rank β In

fact there is then an immediate successor (gprime) of (g)13n in T (B) of maximalinvariant (β l) (where l is the largest possible size of a smallest set in a partitionof k into n nonempty sets) We can thus let f((g)13n) = ω(2β + 1) + ql + 1 Wenote that l = kn

Having defined f on the nodes (g)13n we can extend it to nodes (g)13(n m)because from (g)

13(n 1) to (g)13n there is no splitting on T (B) We thus have

f((g)13(n 1)) = ω(2β) + qn minus 2 if n gt k ω(2β + 1) + ql + qn minus 1 if n le k wherekn lt Since we chose the qks to rise quickly we have ql + qn le qk for n lt kThus if k ge 2 then we indeed have that (g)13(k 1) has maximal rank among theimmediate successors of (g) and indeed we assigned it rank f((g)) minus 1 so f iscontinuous at (g) If k = 1 then for all n f((g)13(n 1)) = ω(2β) + qn minus 2 andqn rarr infin so again f is continuous at (g) as required

Corollary 442 (ACA0) Suppose that A B are ranked Boolean algebras ThenA sim= B iff T (A)infin sim= T (B)infin and A B iff T (A)infin T (B)infin



Note that this yields another proof that ATR0 implies the various statementsfor superatomic Boolean algebras we deduce them from RK(SABA) and the cor-responding statements for fat trees


Proposition 443 (RCA0) exist-EMB(SABA) implies ACA0

Proof This is immediate (same as Proposition 328)

Proposition 444 (RCA0) COMP(SABA) implies ACA0

Proof Let f N rarr N be a one-to-one function For each n isin N let Bn be a finiteBoolean algebra such that if n is not in the range of f it has only one atom andif n = f(m) it has m many atoms Let B =

oplusnisinN

Bn and let Bprime be a rankedBoolean algebra of invariant (ω 2) First we observe that Bprime B Bprime has twoelements with meet 0Bprime and infinitely many atoms below but B does not SoCOMP(SABA) implies B Bprime Let g be such an embedding For at most two(actually one) values of n we may have rk(g(1Bn

)) = 1 For every other n we haveinv(g(1Bn

)) = (0 kn) As in the proof for trees we observe that n is in the rangeof f iff it is in the range of f kn

441 WQO As in subsection 331 we go via RCA2 As mentioned in that sub-section we make use of some ideas of Shorersquos proof that WQO(On) implies ACA0

[Sho93 Theorems 217 and 31]

Proposition 445 (RCA2) WQO(SABA) implies ACA0

Proof Again fix an enumeration of 0prime Let βnt = ω + 1 if t is the kth true stage of

the enumeration for some k gt n and finite otherwise [Set 1 ltlowast 2 ltlowast 3 ltlowast middot middot middot ltlowast 0At stage s gt t determine that s isin βn

t if at s t appears to be a true stage and thereare n other stages before t which also appear to be true at s If t is not true theneventually this will be found out If t is true then already at t we know all the truestages lt t so if there are fewer than n true stages lt t then no s gt t is in βn

t ]Let αn =

sumtisinN

βnt and let Bn = Int(αn) Each Bn is superatomic (to see this

quickly note that Bn =oplus

t Int(βnt ) and see Lemma 421) Suppose that n lt m

and that f Bn rarr Bm is an embeddingBy Σ0

2-induction there are more than n many true stages Suppose then thatt is the (n + k)th true stage for some k gt 0 Thus the interval I =

⋃slet βn

s hasexactly k limit points which in Bn means that it is the join of k many elementsbelow each of which there are infinitely many atoms This has to be true of f(I)in Bm Suppose that sup f(I) isin βm

r Then r bounds a true stage larger than t(for example the (m + k)th true stage) The stage u between t and r at which thesmallest number ever to be enumerated between t and r was actually enumeratedis a true stage gt t This process can be iterated to get infinitely many true stagesand thus 0prime

Now we observe that in light of Lemma 331 and the proof of Proposition 332in order to show

Proposition 446 (RCA0) WQO(SABA) implies IΣ2

it is enough to show two things in RCA0 first that if α β are ordinals and α βthen B(α) B(β) (this is Lemma 433) and that if α is an ordinal then B(α + 1)



does not embed into B(α) Armed with these facts the proof follows exactly as itdid for trees The second statement can be shown using

Fact (RCA0) Suppose that A and B are ranked Boolean algebras with resolutions〈Iβ〉βltα and 〈Jβ〉βltα along the same ordinal α (we allow for a final segment ofIβ = A or Jβ = B) Suppose that f A rarr B is an embedding Then for allβ lt α and x isin A if rk(x) ge β then rk(f(x)) ge β This is shown by Π0

1-transfiniteinduction on α

5 Reduced p-groups

Fix a prime number p A p-group is a group in which every element has order apower of p

Convention From now on all groups are abelian

A group G is divisible if for every a isin G and every n isin N there exists b isin Gsuch that nb = a

Definition 51 A group is reduced if it has no divisible subgroup

Fact (ACA0) A p-group G is reduced iff there is no sequence 〈gn〉nisinNof elements

of G such that for all n pgn+1 = gn This is because the subgroup generated bythe gns is the direct limit of Zpn and in each Zpn one can divide by numbers notdivisible by p

Again reduced groups are the ldquowell-founded partrdquo of the collection of groupsand it takes Π1

1-comprehension to weed out this part A classic result is the followingtheorem of Friedman Simpson and Smith

Theorem 52 ([FSS83]) The statement ldquoevery group is the direct product of areduced group and a divisible grouprdquo is equivalent to Π1

1-CA0 over RCA0

As our notion of embedding we take the usual notion of group embedding(one-to-one homomorphism)

51 Ranked p-groups We define rank functions for reduced p-groups Let α bean ordinal and G a p-group A partial Ulm resolution of G along α is a sequence ofsubgroups 〈Gβ〉βltα such that G0 = G if β + 1 lt α then Gβ+1 = pg g isin Gβand for limit λ lt α Gλ =

⋂βltλ Gβ (we sometimes write pβG for Gβ) An Ulm

resolution of G is a partial Ulm resolution of G along some ordinal α + 1 such thatGα = 0 and for all β lt α Gβ = 0 We call such an α the length of G

Notation (RCA0) If G is a p-group then G[p] its socle is the subgroup consistingof elements of G of order p

Suppose that 〈Gβ〉βleα is a resolution of a p-group G we define an associatedUlm sequence For each β lt α Gβ[p]Gβ+1[p] is a vector space over Zp we letUG(α) be its dimension The sequence 〈UG(β) β lt α〉 is called the Ulm sequenceof G and it characterizes G up to isomorphism We also define an associated rankfunction for x isin G let rkG(x) be the unique β lt α such that x isin Gβ Gβ+1 (InACA0 if a resolution exists then so does the sequence and the rank function butnot in RCA0)



Definition 53 A p-group G is weakly ranked if it has an Ulm resolution and theassociated rank function exists It is ranked if furthermore the associated Ulmsequence exists

RCA0 is enough to prove that reduced p-groups with the same Ulm sequence areisomorphic [Sim99 Theorem V71] Simpson [Sim99 Lemma V72] uses ACA0 toprove that Ulm sequences are unique up to isomorphisms of ordinals

Lemma 54 (RCA0) Let G be a reduced p-group Let α and αprime be ordinals andsuppose that 〈Gβ〉βltα and 〈Gprime

β〉βltαprime are two resolutions of G Furthermore as-sume that the associated rank functions rk and rkprime exist Then α sim= αprime and theisomorphism commutes with rk rkprime

Proof This is similar (but not identical) to the proof of Lemma 416 By Π01-

transfinite induction on β lt α we show that for all x of rank β Grk(x) = Gprimerk(xprime)

(in particular it follows that if rk(x) = rk(y) = β then rkprime(x) = rkprime(y)) Supposethat the claim is verified up to β There are two options

First suppose that β is a limit ordinal Then Gβ =⋂

γltβ Gγ Let x have rankβ and let βprime = rkprime(x) So far we have a map f β rarr αprime which is defined by takingγ lt β to the unique γprime such that for some (all) y isin G of rk γ rkprime(y) = γprime (soGγ = Gprime

γprime) For all γ lt β we thus have x isin Gprimef(γ) so range f sub βprime Of course

f is order-preserving and since β is a limit ordinal range f has no last elementSuppose that f is not cofinal in βprime that there is some δprime lt βprime such that for allγprime isin range f γprime le δprime But then we have Gβ =

⋂γprimeisinrange f Gprime

γprime sup Gprimeδprime on the other

hand βprime gt δprime implies that there is some y isin Gprimeδprime such that py = x But there is no

such y in Gβ Thus βprime = sup range f is a limit ordinal and Gprimeβprime = Gβ as required

Next suppose that β = γ + 1 Take x isin G of rank β let βprime = rkprime(x) Sincex isin Gγ+1 p|x in Gγ so we can find y isin Gγ such that py = x We cannot havey isin Gβ (or x isin Gβ+1) so rk(y) = γ Let γprime = rkprime(y) By induction Gγ = Gprime

γprime so Gβ = Gprime

γprime+1 But βprime = rkprime(x) = γprime + 1 Why is that Otherwise we haverkprime(x) gt γprime + 1 so x isin Gprime

γprime+2 thus there is some y isin Gprimeγprime+1 such that py = x and

so there is some z isin Gprimeγprime such that pz = y But then z isin Gγ and so x isin Gγ+2

which is false Thus Gβ = Gprimeβprime as required

When we are done with the induction we define f α rarr αprime as we did in the limitstages and get the desired isomorphism Lemma 55 (RCA0) Every ranked p-group is reduced

Proof Suppose G is a p-group which is not reduced Let H be a divisible subgroupof G and let x0 isin H By primitive recursion construct a sequence 〈xi i isin N〉 suchthat for each i pxi+1 = xi Note now that for each i rk(xi+1) lt rk(xi) because ifxi+1 isin Gβ then xi isin Gβ+1 So we have a contradiction because the length of G iswell-founded

The statement that every reduced p-group is weakly ranked is equivalent to ATR0

over RCA0 [Sim99 Theorem V73] It follows that the statement that every reducedp-group is ranked is also equivalent to ATR0 over RCA0 because as mentionedearlier ACA0 is enough to compute the Ulm sequence

Given a group G a set A sube G 0 is independent if for any a1 ak isin A ifn1a1 + middot middot middot + nkak = 0 then n1a1 = middot middot middot = nkak = 0 Let G be a p-group Then if



a1 ak isin G are independent and the order of ai is pni then pn1minus1a1 pnkminus1ak

is an independent subset of G[p] which is a vector space Hence all maximalindependent sets in G have the same cardinality which we denote by r(G) the (forour purposes unfortunately named) rank of G r(G) = r(G[p])

Lemma 56 (ACA0) Suppose that G Gprime are ranked p-groups (with resolutions〈Gβ〉βleα 〈Gprime

βprime〉βprimeleαprime) Suppose that f G rarr Gprime is an embedding Then there is

some embedding h α rarr αprime such that for all β lt α r(Gβ) le r(Gprimeh(β))

(This extends the discussion about embeddings in [Frib])

Proof For β lt α we let h(β) be the maximal βprime lt α such that fldquoGβ sub Gprimeβprime h

is strictly order-preserving because for all β lt α Gβ+1 = pGβ sub Gprimeh(β)+1 Now

h Gβ[p] is a vector space embedding of Gβ[p] into Gprimeh(βprime)[p] so the rank cannot

decrease

Remark 57 Suppose that 〈Gβ〉βleα is the Ulm resolution of a p-group G Thenfor β lt γ le α we have r(Gβ)minus r(Gγ) =

sumδisin[βγ) UG(δ) Thus if α = ε + n where

ε is a limit ordinal then for all k lt n r(Gε+k) = UG(ε + k) + middot middot middot + UG(ε + n minus 1)If β lt ε then there are infinitely many δ isin [β ε) such that UG(δ) gt 0 and sor(Gβ) = ω (See Barwise and Eklof [BE71])

We in fact have a converse for Lemma 56


βprime〉βprimeleαprime) Suppose that there is an embedding h α rarr αprime such that for

all β lt α r(Gβ) le r(Gprimeh(β)) Then G Gprime

Proof See [BE71 Corollary 54] and [Frib]

As before we need to see how ranks correspond to direct sums

Lemma 59 (RCA0) Suppose that 〈Hn〉 is a sequence of reduced p-groups ThenG =

oplusn Hn is reduced If 〈Gβ〉βleα is the Ulm resolution for G then for all n isin N

〈Gβ cap Hn〉βleα is an Ulm resolution of Hn (we may need to trim the end of thesequence though)

Proposition 510 (ACA0) RK(R-p-G) implies exist-EMB(R-p-G)

Proof This is similar to what we did before Let 〈Hn〉 be a sequence of reducedp-groups Let G =

oplusn Hn and get a resolution of G we get the induced resolutions

of the Hns uniformly Again comparability of the ordinals in question is equivalentto their position in the length of G we can check ranks of tail-ends to see if thecondition in Lemmas 56 58 holds

Similarly

Proposition 511 (ACA0) RK(R-p-G) implies exist-ISO(R-p-G)

Proof Note that we need ACA0 to check not only equality of lengths but also ofthe Ulm function along the length



52 Reductions

521 Ordinals to groups

Definition 512 Given a tree T we let G(T ) be the (abelian) group generatedfreely by the elements of T modulo the relations 〈〉 = 0 and pτ = σ whenever τ isan immediate successor of σ on T Given an ordinal α we let G(α) = G(T (α))

Despite the presentation as generators relations the group G(T ) is computablefrom T For more information on the reduction G(T ) as for example how tocompute its Ulm Sequence see [Bar95]

The following is proved in [Frib]

Lemma 513 (RCA0) For any tree T T is well-founded iff G(T ) is reduced

Also

Lemma 514 (RCA0) For any ordinal α G(α) is a weakly ranked p-group of lengthα

As usual in ACA0 we can also prove that G(α) is ranked

Proof following [Frib] Given β lt α let Gβ be the set of elements of G(α) of theform n0σ0 + middot middot middot + nkσk such that for all i le k the last element of σi is at least βWe claim that 〈Gβ β lt α〉 is an Ulm resolution for G(α) Clearly for λ a limitordinal Gλ =

⋂βltλ Gβ Now consider x = n0σ0 + middot middot middot + nkσk If x isin Gβ+1 then

y = n0σ0β + middot middot middot + nkσk

β isin Gβ and py = x Conversely if x isin Gβ+1 thenfor some σi the last element of σi is at most β So there is no y isin Gβ such thatpy = x Thus 〈Gβ〉βleα is indeed an Ulm resolution of G(α) It is also easy to findthe rank of any element

Lemma 515 (RCA0) Given ordinals α and β we have that α sim= β lArrrArr G(α) sim=G(β)

Proof Clearly if α sim= β then G(α) sim= G(β) Suppose now that G(α) sim= G(β) Bythe previous lemma G(α) and G(β) are both weakly ranked and have lengths α andβ By Lemma 54 we have that α sim= β

Corollary 516 (RCA0) exist-ISO(R-p-G) implies ATR0 Therefore RK(R-p-G) andexist-ISO(R-p-G) are equivalent to ATR0

Lemma 517 (ACA0) Given ordinals α and β we have that α β lArrrArr G(α) G(β)

Proof If G(α) G(β) then by Lemma 58 α β If α β then we can directlyconstruct an embedding of G(α) into G(β)

Corollary 518 (ACA0) exist-EMB(R-p-G) implies ATR0

522 Groups to trees Given a p-group G we let T (G) (essentially) consist of theelements of G we declare that 0G corresponds to 〈〉 and x isin G of order pn isidentified with the sequence 〈pnminus1x px x〉 It is immediate that G is reducediff T (G) is well-founded if G is ranked then so is T (G) (the rank of every nonzerox on T (G) is its rank in G the rank of the tree is the length of G) As we noticedhowever because of the intricate structure of the equimorphism classes of reducedp-groups this operation can preserve neither nonisomorphism nor nonembedding




Proposition 519 (RCA0) exist-EMB(R-p-G) implies ACA0 and hence it is equivalentto ATR0

Proof Let ϕ be a Σ01-formula For each n construct a p-group Gn by letting

Gn = Zp if notϕ(n) and Gn = Zp2 if ϕ(n) Then n ϕ(n) = n Zp2 Gn

We next show that WQO(R-p-G) implies ACA0 As before we go through Σ02-

induction Let T and the sequence 〈Tn〉 be as in subsection 331 and let Gn =G(Tn) Again assuming that 0prime does not exist by the results of Friedman quotedearlier (Lemma 513) each Gn is reduced

Suppose n lt m and that g Gn rarr Gm is an embedding Suppose that σ isin Tn

is a true string Considered as an element of Gn we write g(σ) in normal form assumiltk miσi where σi isin Tm and mi isin Zp We claim that every σi is true Suppose

that some σj is not true let r be the height of the (finite) tree Tm[σj ] By Σ02-

induction there is some true τ sup σ on Tn which is sufficiently long so that psτ = σwhere s gt rminus|σj | Then psg(τ ) = g(σ) Writing g(τ ) in normal form as

sumiltkprime niτi

and multiplying by ps we get

g(σ) =sum

iltk

mi σi =sum

iltkprime

ni τ primei

where τ primei is τi with the last s bits chopped off Thus the set of the τ prime

is equals theset of the σis which shows that some τi is an extension of σj of length gt r Thisis impossible

By the same kind of calculation we see that if σ0 σ1 isin Tn are true then eachτ appearing in the normal form of g(σ0) is properly extended by some τ prime whichappears in g(σ1) This shows that if σ isin Tn is true then via g we can obtain sometrue τ isin Tm of length at least |σ| This allows us to iterate and get 0prime

Next we see that Σ02-induction follows from WQO(R-p-G) This follows the proof

of Lemma 332 As for Boolean algebras all we really need is that in RCA0(1) If α β are ordinals and α β then G(α) G(β)(2) For any ordinal α G(α + 1) does not embed into G(α)

As for Boolean algebras (subsection 441) the first follows from a direct construc-tion and the second follows from an analogue of fact 441 (with the same proof)Thus

Proposition 520 (RCA0) WQO(R-p-G) implies ACA0

6 Scattered and compact spaces

In this section we introduce the class of very countable topological spaces Un-fortunately the well-founded part (the collection of scattered spaces) of our classis not very well-behaved We thus leave open the analysis of the class of scatteredspaces and concentrate on compact spaces which turn out to be metrizable Thisallows us to refer to a rich body of research on metric spaces in reverse mathematics

Remark 61 We do not give a reduction from topological spaces to other classesIn fact we do not know whether such computable embeddings exist There areTuring reductions from compact spaces to well-founded trees and ordinals whichpreserve embedding nonembedding isomorphism and nonisomorphism however



they make use of a particular listing of the points of the space or of its basic opensets

The most natural reduction from compact spaces is to the class of superatomicBoolean algebras ie Stone duality All facts about Stone duality can in factbe proved in ATR0 (including the fact that the corresponding Boolean algebra iscountable) however this is not a continuous operation so we do not consider it inthis paper

61 Definitions

Definition 62 A very countable topological space is a set X sub N equipped witha (countable) collection of subsets OX which are a basis for a topology on X (iefor every finite subset F of OX and every x isin

⋂F there exists U isin OX such that

x isin U sube⋂

F )We assume all the usual topological notions (see for example [Mun00]) So for

instance V sube X is an open set (or an OX-open set) if forallx isin V existU isin OX(x isin V sube U)Two topologies OX and Oprime

X on a same set X are equivalent if the OX -open sets areexactly the Oprime

X -open sets Note that up to equivalence we can assume that OX isalways closed under finite intersections since closing up under finite intersectionsis a computable operation

As isomorphism we use homeomorphisms A one-to-one map f X rarr Y is bi-continuous if it is continuous and its inverse is continuous as well (for formalizationin RCA0 we do not assume that a map necessarily has a range the notion ofcontinuity still makes sense because the range is definable) An embedding of aspace X into a space Y is a one-to-one bi-continuous function An embeddingf X rarr Y is a homeomorphism of X onto its perhaps nonexistent range We notethat the standard definition of the subspace topology makes sense in this setting

All spaces we deal with are very countable and so we drop this prefix Alsounless otherwise stated all spaces are Hausdorff (That is for every x y isin X thereexist disjoint U V isin OX such that x isin U y isin V ) In Hausdorff spaces we can usefamiliar notions such as converging sequences and limit points So when we sayspace we mean very countable Hausdorff topological space

We note that very countable topological spaces are just countable second count-able spaces The reader familiar with these concepts should note that when a spaceis countable the notions of first countable (N1) and second countable (N2) coincideWe deal with very countable topological spaces because they are the ones that canbe easily encoded in Second Order Arithmetic as a set of natural numbers (ratherthan a class of reals)

Example (RCA0) Let L be a linear ordering There is a natural topology on Lthe order topology which makes L a very countable space OL is the set of openintervals of L defined by endpoints in A cup minusinfininfin

We observe that if B is a subordering of A then the order topology on B isequivalent to the subspace topology

A ldquowell-foundedrdquo topological space is called scattered

Definition 63 Let X be a space We say that y isin Y is isolated if y isin OY Aset Y sub X is dense in itself if as a subspace Y has no isolated points A space isscattered if it contains no subset which is dense in itself



62 Metrizable spaces A metric space consists of a set M sub N and a sequence〈r(a b)〉abisinM of real numbers (real numbers in the sense of [Sim99 Chapter II]quickly converging Cauchy sequences of rationals) satisfying the classical propertiesof a metric

Recall that a metric on a set M induces a topology on it OM = Bxr x isinM r isin Q (Of course we can then close OM under finite intersections) Atopological space is metrizable if there is a metric on X consistent with its topologyThat is a metric such that the topology it induces is equivalent to the topology onX

Example (RCA0) If α is an ordinal then there is a canonical embedding of α intothe interval (0 1) see [FH91] This embedding is bi-continuous with respect to theorder topology on α which shows that α as a topological space is metrizable

A topological space X is normal if it is Hausdorff and for every pair of disjointclosed sets C D sub X there exist disjoint open sets U and V such that C sube U andD sube V X is regular if the condition above holds when C is a singleton

Lemma 64 A space X is regular iff for all x isin X and all open neighborhoods Uof x there is some neighborhood V of X such that V sub U

The proof in [Mun00 Theorem 311] goes through in ACA0We note that being an open subset of X is an arithmetic property Similarly

the relation x isin A is arithmetic as the closure of A sub X is the collection of pointswhich do not have basic open neighborhoods disjoint from A As a corollary of theprevious lemma we notice that regularity is arithmetically definable as a space Xis regular iff for all x isin X and all basic open neighborhoods U of x there is somebasic neighborhood V of x such that V sub U Furthermore we note that if X isregular then the function taking some x isin X and a basic neighborhood U of x tothe least (according to some fixed enumeration) basic neighborhood V of x suchthat V sub U is arithmetically definable It is not ldquotopologicalrdquo as it depends on theenumeration of OX - so does not respect homeomorphism Of course we view thestructure X as equipped with some enumeration of OX so the function is indeeddefinable from X

Similarly the function taking some closed A sub X and x isin X A to the leastbasic neighborhood of x disjoint from B is arithmetically definable

Lemma 65 (ACA0) Every regular space is normal

Proof Essentially the proof is the standard one given for example in [Mun00Theorem 321] All we need is to note that when for each x isin A we choose abasic open neighborhood of x whose closure is disjoint from B we can do thatarithmetically and so the resulting cover exists

The proof in fact yields more if X is regular then there is an arithmeticallydefinable function which takes disjoint closed A B sub X to a pair of open subsetsof X which separate A and B Or equivalently given some closed A and openB sup A we can get in an arithmetical way some open C sup A such that C sub B

We need to refine even further Let X be a normal space Fix some enumeration〈Un〉nisinN

of OX Suppose that A sub X is open An open presentation of A is a setN sub N such that A =

⋃nisinN Un A closed presentation of a closed set B sub X is an

open presentation of X B Among all open presentations of some open A sub X



there is one that is maximal it is of course n Un sub A Given some open setA sub X ACA0 ensures that some presentation of A in fact its maximal one exists

ACA0 ensures the existence of sets such as (x n) x isin Un (n m) Un subUm and functions such as the one taking n to the maximal closed presentationof Un and (n m) to the maximal closed presentation of Un cap Um When we fixthese sets and functions as oracle we can by the proof of Lemma 65 effectivelyconstruct given some closed presentations of disjoint A B sub X open sets U Vseparating A and B furthermore we can effectively construct open presentationsof U and V This allows us to iterate the process of finding separators which showsthat the proof of Urysohnrsquos Lemma goes through in ACA0

Lemma 66 (ACA0) (Urysohnrsquos Lemma [Mun00 Theorem 331]) If X is a normalspace and A B are disjoint closed subsets of X then there is some continuousf X rarr [0 1] such that A sub fminus10 and B sub fminus11Remark 67 We may assume that range f sub Q For given f we can use the stan-dard ldquoforthrdquo argument to get an order-preserving hence continuous g range f rarrQ ACA0 ensures that lt range f exists

We also remark that f is obtained effectively given the discussed oracle uniformlyin (closed presentations of) A and B This uniformity allows us to see that if X isregular then there is a countable collection of functions f X rarr [0 1] such that forany x isin X and any neighborhood U of x some f in the collection is positive at xand vanishes outside U The rest of the proof of the Urysohn metrization theoremgoes through in ACA0

Theorem 68 (ACA0) (Urysohn metrization theorem [Mun00 Theorem 341]) LetX be a space The following are equivalent

(1) X is regular(2) X is normal(3) X is metrizable

Moreover given a sequence of regular topological spaces there is a sequence ofmetrics for them

Example (ACA0) The order topology of every linear ordering is regular and hencemetrizable

63 Compact spaces

Definition 69 A space X is compact if every open covering of X which consistsof basic open sets contains a finite subcovering That is if for every sequence ofbasic open sets 〈Un〉nisinN

such that X sube⋃

nisinNUn there exists a finite F sub N such

that X sube⋃

nisinF Un

We note that when working in ZFC the definition of compactness requires con-sideration of uncountable open coverings But if a space is countable then everyopen covering has a countable subcovering Furthermore if a space is very count-able then from an arbitrary countable open covering we can find a refinementwhich is both countable and consists of basic open sets So when dealing withvery countable spaces the definition of compactness given above is equivalent tothe usual one

Definition 610 A space X is sequentially compact if every infinite sequence ofelements of X has a converging subsequence



The trick of having every infinite Σ01-class contain an infinite set yields the fol-

lowing

Lemma 611 (RCA0) A space X is sequentially compact iff every infinite subsetof X has a limit point in X

The standard proofs of equivalence formalized yield the following

Lemma 612 (RCA0) Every sequentially compact space is compact

Lemma 613 (ACA0) Every compact space is sequentially compact

We observe that when working in ZFC with countable compact Hausdorffspaces the condition of very countability comes for free

Observation (ZFC) Every countable compact Hausdroff topological space issecond countable and hence very countable

Proof Suppose that X is not second countable Then it is not first countableeither That means that there exists an x isin X which has no countable basis ofopen neighborhoods ie such that for every decreasing sequence 〈Un〉nisinN

of openneighborhoods of x there exists an open neighborhood V of x such that for no nUn sube V

Let 〈xn〉nisinω be an enumeration of X x We will construct a subsequence〈xnk

〉kisinω with no converging subsequence contradicting the compactness of XFirst using the fact that X is Hausdorff construct two sequences of open sets〈Un〉nisinω and 〈Vn〉nisinω such that for every n Un and Vn are disjoint neighborhoodsof x and xn respectively Un+1 Un Now there is an open set W containing x suchthat for no n Un sube W Define 〈xnk

〉kisinω as follows Let xn0 isin U0 W given xnk

let m gt nk be such that xnkisin Um and let xnk+1 isin Um W Now x is not a limit

of any subsequence of 〈xnk〉kisinω because W is a neighborhood of x which contains

no point in that sequence Also any point xn is not a limit of any subsequenceof 〈xnk

〉kisinω because Vn is a neighborhood of xn which contains no point xnkfor

nk gt n

Observation The above proof can be carried through in a subsystem of second-order arithmetic with sufficiently much choice provided that it is meaningful thatis when the given space is a definable class

We now see that compact spaces are nice and well-founded For the first thestandard proof will do

Lemma 614 (ACA0) Every compact space is normal

Lemma 615 (ACA0) Every compact space is scattered

Proof Suppose that X is a topological space and suppose that Y sub X is dense initself We will construct a sequence 〈yn〉nisinω sube Y with no convergent subsequenceLet 〈xn〉nisinω be an enumeration of X Let y0 be such that there exists disjointbasic open sets U0 and V0 around y0 and x0 respectively Suppose we have alreadydefined yn and Un and yn isin Un Let yn+1 isin Un be any point different from xn+1which exists because Y is dense in itself Let Un+1 sube Un and Vn+1 be basic openneighborhoods of yn+1 and xn+1 respectively Now since for every n xn isin Vn andforallm ge n (ym isin Vn) 〈yn〉 has no convergent subsequence



We shall need the following basic facts(1) If X is compact and Y sub X is closed then Y is compact (in the subspace

topology)(2) If Y sub X is compact and X is Hausdorff then Y is closed in X

Standard proofs go through in ACA0 so we get (remembering that all spaces areHausdorff)

Fact (ACA0) If X is compact and f X rarr Y is one-to-one and continuous then fis closed (so f is an embedding)

631 Spaces of well-orderings In the following by complete spaces we of coursemean perhaps uncountable definable spaces

Lemma 616 (ATR0) [FH91] Every countable closed totally bounded subset of acomplete separable metric space is homeomorphic to the canonical metric space ofsome well-ordered set

Lemma 617 (ACA0) Let X be a topological space The following are equivalent(1) X is compact(2) X is homeomorphic to a countable closed totally bounded subset of a com-

plete separable metric space

Proof If X is compact then we can consider it as a metric space We let X be thecompletion of X (see [Sim99 Section II5]) This is a complete separable metricspace Since X is compact it is closed in X and is also totally bounded

For the other direction see [Mun00 Theorem 451] Thus every compact space is homeomorphic to the order topology of some ordi-

nal

Lemma 618 (RCA0) Let α be an ordinal Then α + 1 is a compact space

This can be derived in ACA0 from Theorems 22 and 23 of [FH91]

Proof Let 〈Un〉 be a sequence of basic open sets which cover α + 1 The pointis that from n we can get the pair (an bn) defining Un Thus if 〈Un〉 does nothave a finite subcover we can inductively choose a descending sequence 〈ck〉 inα as follows together with 〈ck〉 we find a sequence 〈Unk

〉 such that for each k(ckinfin) sub

⋃llek Unl

Given ck we let Unk+1 be some open set on the list whichcontains ck and we let ck+1 = ank+1

We can now define the reduction from ordinals to compact spaces

Definition 619 Let α be an ordinal and n isin N The space C(α n) is thetopological space given by the order topology on ωα middotn+1 We let C(α) = C(α 1)

64 Ranked spaces The Cantor-Bendixon derivative X prime of a space X is thecollection of limit points of X (ie X with its isolated points removed) Let αbe an ordinal a partial Cantor-Bendixon resolution of X along α is a sequence ofsubspaces 〈Xβ〉βltα such that X0 = X if β + 1 lt α then Xβ+1 = X prime

β and forthe limit λ lt α Xλ =

⋂βltλ Xβ A Cantor-Bendixon resolution of X is a partial

Cantor-Bendixon resolution of X along some ordinal α + 1 such that Xα = empty butfor β lt α Xβ = empty If there is a resolution of X along α + 1 then we let rk(X) = α



(which is also called the length of X) If α is a successor ordinal then the degreeof X is the number of points in Xαminus1 if α is a limit ordinal we let deg(X) = 0We define inv(X) = (rk(X) deg(X)) We also define an associated rank functionfor x isin X we let rk(x) be the unique β lt α such that x isin Xβ Xβ+1

Definition 620 A space X is ranked if it has a Cantor-Bendixon resolution andthe associated rank function exists

As usual in RCA0 we can show that any two rankings of a ranked space X areisomorphic See for example Lemma 416

Lemma 621 (ATR0) Every scattered space is ranked

Proof Friedman [Fria] proved this result for countable metric spaces The sameproof works for topological spaces Lemma 622 (RCA0) Every ranked space is scattered

Proof Suppose that Y sub X is dense in itself Let 〈Xβ〉βltα be any partial resolutionof X Then by Π0

1-transfinite induction on β lt α we can show that Y sub Xβ forevery β lt α Thus 〈Xβ〉 cannot be a full resolution of X Lemma 623 (RCA0) Let α be an ordinal and n isin N Then C(α n) is scatteredindeed it is ranked and its invariant is (α + 1 n)

Of course in RCA0 C(α n) is compact

Proof Let X = C(α n) Given x =sum

iltk ωβi middotni isin X we let rk(x) = minβi i ltk For β le α+1 let Xβ = x isin X rk(x) ge β We claim that 〈Xβ β le α+1〉 isa Cantor-Bendixon resolution of X and rk is a rank function for X It is clear thatwhen γ is a limit ordinal Xγ =

⋂δltγ Xδ We then have to prove that for every

γ le α Xγ+1 = X primeγ This follows from the fact that Xγ = ωγ middot δ δ le ωαprime middot n

and Xγ+1 = ωγ middot δ δ le ωαprime middot n and δ is a limit ordinal where αprime is such thatγ +αprime = α (note that we do not assume that we can regard αprime as an initial segmentof α)

It follows that rk(X) = α+1 and since Xα = ωαmiddoti 0 lt i le n deg(X) = n The importance of the Cantor-Bendixon invariant is that it classifies compact

spaces up to isomorphism and is also compatible with the embedding relationFor the case of countable metric spaces Friedman essentially proved the followinglemma

Lemma 624 (ACA0) ([Fria]) Let X and Y be ranked countable metric spaces withinvariants 〈α n〉 and 〈β m〉 respectively Then there is a one-to-one continuousfunction f X rarr Y if and only if 〈α n〉 lelex 〈β m〉

Recalling fact 63 we get

Corollary 625 (ACA0) Let X and Y be ranked compact spaces Then X Y iffinv(X) lelex inv(Y )

As in earlier sections we will want to get uniform rankings of a sequence ofspaces

Lemma 626 (ACA0) Assume that every compact space is ranked Let 〈Xn〉 be asequence of compact spaces Then there is an ordinal α and a sequence of functionsfn Xn rarr α such that each fn is a rank function for Xn



Proof Let Y be the disjoint union of the Xns and let X be a simple version ofthe 1-point compactification of Y The basic open subsets of X are the basic opensubsets of Y together with the sets X Xn for n isin N It is straightforward to checkthat X is compact and that each Xn is an open subset of X Thus a ranking of Xgives uniform rankings of all the Xns (see Friedman [Fria] to see that if A sub B isopen in B and 〈Bγ〉 is a resolution of B then 〈A cap Bγ〉 is a resolution of A) Corollary 627 (ACA0) RK(CCS) implies exist-EMB(CCS) COMP(CCS) andWQO(CCS)

Corollary 628 (ATR0) Every compact space X is homeomorphic to C(α n)where (α + 1 n) = inv(X)

Proof By Lemmas 616 and 617 X is homeomorphic to the canonical metric spaceof some ordinal β (Note that β has to be a successor ordinal because otherwiseX would not be compact) Using the Cantor normal form write β as

β = ωα0 middot n0 + ωα1 middot n1 + + ωαk middot nk

where α0 gt α1 gt gt αk We can write β as

(ωα0 middot n0 + 1) + (ωα1 middot n1 + 1) + + (ωαkprime middot nprimekprime + 1)

It is not hard to prove that β is homeomorphic to

(ωαkprime middot nprimekprime + 1) + + (ωα1 middot n1 + 1) + (ωα0 middot n0 + 1)

which as and ordinal is isomorphic to ωα0 middot n0 + 1 Finally (α + 1 n) = inv(X) =inv(ωα0 middot n0 + 1) = (α0 + 1 n0) so α0 = α and n0 = n Corollary 629 (ATR0) Let X and Y be compact spaces with invariants 〈α n〉and 〈β m〉 respectively Then X and Y are homeomorphic if and only if 〈α n〉 =〈β m〉Corollary 630 ATR0 implies the statements exist-ISO(CCS) and EQU = ISO(CCS)

65 Reversals To get reversals we need to apply some of the aforementionedresults in weaker systems The next lemma follows from Lemma 623

Lemma 631 (RCA0) Let α and β be ordinals Then α sim= β iff C(α) and C(β)are homeomorphic

Corollary 632 (RCA0) exist-ISO(CCS) is equivalent to ATR0

The next lemma follows from Lemma 623 and Corollary 625

Lemma 633 (ACA0) Let α and β be ordinals Then C(α) embeds in C(β) if andonly if α β

Corollary 634 (ACA0) Each of exist-EMB(CCS) COMP(CCS) EQU = ISO(CCS)and WQO(CCS) is equivalent to ATR0

To get the EW -reduction we first observe the following Suppose that L is anill-founded linear ordering let 〈an〉 be an infinite descending sequence in L Wecan construct a copy of 1 + Q inside ωL by considering all elements of the formωa1n1+ωa2n2+middot middot middot+ωaknk for k nl isin N Thus ωL is not scattered As a conclusionwe see that for all linear orderings L if L is a well-ordering then C(L) is compactand if L is ill-founded then C(L) is not scattered and so not compact All thiscan be done in RCA0




Proposition 635 (RCA0) exist-EMB(CCS) implies ACA0

Proof See Proposition 328 or 519 Proposition 636 (RCA0) RK(CCS) implies ACA0

Proof This is immediate given Lemma 626 We construct a sequence 〈Xn〉 ofcompact spaces and points xn isin Xn such that if n isin 0prime then xn is a limit point inXn and if n isin 0prime then Xn = xn Then from a uniform ranking of the Xns wecan uniformly get the rank of each xn (in Xn) and thus get 0prime Proposition 637 (RCA0) COMP(CCS) implies ACA0

Proof We again show that 0prime exists by showing how to enumerate infinitely manytrue stages

As in the proof of Proposition 445 we construct a sequence of ordinals 〈αs〉sisinN

such that if s is a true stage then αs is a canonical copy of ω + 1 and otherwiseαs is a copy of some n lt ω (where we can tell which is the last element and whatis the place of the other elements) We let α =

sumsisinN

αsIt is easy to see that α is an ordinal (from a decreasing sequence in α we can

construct either a decreasing sequence of s isin N or a decreasing sequence in someαs) The limit points of α are exactly those last elements of αs where s is a truestage

Let X = α middot 2 + 1 with the order topology Let Y be a canonical copy of ω2 + 1There cannot be an embedding of X into Y This is because for any embeddingthe image of a limit point is a limit point and the image of a limit of limit pointsis also a limit of limit points of which X has two but Y only one

By COMP(CCS) there is an embedding g of Y into X Write X = α0 + α1 + 1(αi is a copy of α) and let A be the class of limit points of X For some i lt 2Bi = Acapgminus1αi is infinite For such i g Bi allows us to enumerate infinitely manylimit points of α and so infinitely many true stages Proposition 638 (RCA0) EQU = ISO(CCS) implies ATR0

Proof Let α and β be ordinals we will prove that they are comparable Note thatδ = α + β + α + and γ = β + α + β + α + are equimorphic via continuousembeddings So ωδ +1 and ωγ +1 are also equimorphic by continuous embeddingsand hence equimorphic as compact spaces By EQU = ISO(CCS) we have that theyare homeomorphic Then by Lemma 631 δ and γ are isomorphic It follows thatα and β are comparable Proposition 639 (RCA0) WQO(CCS) implies ACA0

Proof As usual we first work over RCA2We define βn

t and αn in a similar way to what is done in the proof of Proposition445 in this case βn

t = ω + 1 exactly when t is the kth true stage for some k gt 2nαn =

sumt βn

t Let Xn = αn + 1 with the order topology every Xn is compactSuppose that n lt m and that f is an embedding of Xn into Xm

Let tk be the (2n+k)th true stage For all t let at = max βnt and let bt = max βm

t For all k gt 0 atk

is a limit point of Xn Consider f(at1) and f(at2) At least one ofthem is in αm (that is it is not the last limit point we added to make Xm compact)and in fact it has to be btk

for some k gt 2 say for example k = 7 From t7 we



can find t3 t4 t6 From f(at1) f(at2) f(at7) at least six are in αm and arebtk

s for distinct k gt 2 thus at least one must be btkfor k gt 7 Now the process

repeats to get all tkThe rest (getting IΣ2 from WQO(CCS)) is identical to the proof in all the pre-

vious sections That is we verify in RCA0 that if α β then C(α) C(β) andthat the analog of fact 441 holds for compact spaces and so that C(α + 1) doesnot embed into C(α)

7 Up to equimorphism hyperarithmetic is recursive

In this section we prove the theorems in subsection 16 As we mentioned thefirst two are not difficult They rely on the fact already mentioned in [AK00]for Boolean algebras and groups that if X is a hyperarithmetic structure in thewell-founded part of any of the classes we considered its rank is computable Thisis simple we give a general proof for each class X we considered we showedthat ATR0 suffices to prove that each well-founded structure is ranked By [Sim99Corollary VII212] we know that there is a β-model M of ATR0 which consistsof hyper-low sets Each hyperarithmetic structure X in X is in M a rank for Xin the sense of M exists in M and since M is Σ1

1-correct this rank is really anordinal

Since the invariant of a compact space determines its isomorphism type we getTheorem 110 immediately This also yields the result 18 for superatomic Booleanalgebras mentioned in [AK00] We also know that the rank of a well-foundedtree determines its equimorphism type so we get that every hyperarithmetic well-founded tree is equimorphic with a recursive one

Theorem 19 follows because we know that if B is a Boolean algebra which is notsuperatomic then it contains a copy of the atomless Boolean algebra into whichevery countable Boolean algebra can be embedded and if T is an ill-founded treethen every countable tree can be embedded into T

We turn to the third theorem

Proof of Theorem 111 Let G be a hyperarithmetic p-group If G is reduced then(see [AK00 Theorem 817]) it has some length α + n lt ωCK

1 where α is a limitordinal and n lt ω By [Bar95 Proposition 43] and [BE71 Theorem 41] there isa recursive group H of length α + n such that for all β lt α UH(β) = infin and forall m le n UH(α + m) = UG(α + m) From Lemma 58 and Remark 57 we obtainthat H and G are equimorphic

Suppose now that G is not reduced It can be written as a sum Gd + Grwhere Gd is divisible and Gr is reduced (see [Kap69 Theorem 3]) Every countabledivisible p-group is of the form Z(pinfin)m for some m le ω (see [Kap69 Theorem4]) and hence has a recursive copy Again by [AK00 Theorem 817] Gr has somelength α le ωCK

1 If Gr has length α lt ωCK1 by the previous argument it is

equimorphic to a recursive group and hence G = Gd + Gr is too Suppose nowthat α = ωCK

1 We claim that then Gdsim= Z(pinfin)ω and hence G is equimorphic to

Z(pinfin)ω (Note that any countable p-group embeds in Z(pinfin)ω) Suppose insteadtoward a contradiction that Gd

sim= Z(pinfin)n for some n lt ω Note that fromRemark 57 we get that for all β lt ωCK

1 r(pβG) = infin Consider the partialordering P whose elements are n+1 tuples of independent elements of G and suchthat 〈x0 xn〉 le 〈xprime

0 xprimen〉 iff there exists some k isin N such that for every

i le n pkxi = xprimei We claim that P is well-founded and has rank le ωCK

1 This



will be a contradiction because P is hyperarithmetic We prove this by defining arank function on P Given 〈x0 xn〉 isin T first write each xi as yi + zi whereyi isin Gr and zi isin Gd and then let g(〈x0 xn〉) = minrkGr

(yi) i le nwhere rkGr

(0) = infin We claim that g P rarr ωCK1 is a rank function First we

observe that for no x isin P g(x) = infin If g(〈x0 xn〉) = 0 then for all i le nyi = 0 and hence xi = zi isin Gd but this cannot be the case because sincer(Gd) = n x0 xn cannot be an independent set Second we observe that if〈x0 xn〉 lt 〈xprime

0 xprimen〉 then g(〈x0 xn〉) lt g(〈xprime

0 xprimen〉) This is because

if g(〈x0 xn〉) = rkGr(xi0) then g(〈xprime

0 xprimen〉) le rkGr

(xprimei0

) lt rkGr(xi0) Last

we show that if β lt g(〈x0 xn〉) there exists some 〈xprime0 x

primen〉 lt 〈x0 xn〉

such that g(〈xprime0 x

primen〉) ge β By definition of rkGr

for each i le n there existsxprime

i such that xprimeip = xi and rkGr

(xprimei) ge β Clearly 〈xprime

0 xprimen〉 lt 〈x0 xn〉 and

g(〈xprime0 x

primen〉) ge β We still have to prove that 〈xprime

0 xprimen〉 isin P Suppose thatsum

ilen mixprimei = 0 Then

sumilen mixi = p

sumilen mix

primei = 0 and hence mixi = 0 for

every i This implies that p|mi for every i Thensum

ilen(mip)xi =sum

ilen mixprimei = 0

and hence mixprimei = (mip)xi = 0 for every i We have proved that 〈xprime

0 xprimen〉 is

an independent set and hence belongs to P The fact that P has rank ωCK1 follows

from the fact that for all β lt ωCK1 r(pβG) = infin

References

[AK00] C J Ash and J Knight Computable structures and the hyperarithmetical hierar-chy Studies in Logic and the Foundations of Mathematics vol 144 North-HollandPublishing Co Amsterdam 2000 MR1767842 (2001k03090)

[Bar95] Ewan J Barker Back and forth relations for reduced abelian p-groups Ann PureAppl Logic 75 (1995) no 3 223ndash249 MR1355134 (96j03066)

[BE71] Jon Barwise and Paul Eklof Infinitary properties of abelian torsion groups AnnMath Logic 2 (19701971) no 1 25ndash68 MR0279180 (434906)

[Cal04] Wesley Calvert The isomorphism problem for classes of computable fields ArchMath Logic 43 (2004) no 3 327ndash336 MR2052886 (2005b03103)

[CCKM04] Wesley Calvert D Cummins Julia F Knight and Sara Miller Comparing classes offinite structures Algebra and Logic 43 (2004) 374ndash392 MR2135387 (2006e03049)

[CG01] Riccardo Camerlo and Su Gao The completeness of the isomorphism relation forcountable Boolean algebras Trans Amer Math Soc 353 (2001) no 2 491ndash518(electronic) MR1804507 (2001k03097)

[CMS04] Peter Cholak Alberto Marcone and Reed Solomon Reverse mathematics and theequivalence of definitions for well and better quasi-orders J Symbolic Logic 69 (2004)no 3 683ndash712 MR2078917 (2005e03020)

[FH90] Harvey M Friedman and Jeffry L Hirst Weak comparability of well orderings andreverse mathematics Ann Pure Appl Logic 47 (1990) no 1 11ndash29 MR1050559

(91b03100)[FH91] Reverse mathematics and homeomorphic embeddings Ann Pure Appl Logic

54 (1991) no 3 229ndash253 MR1133006 (92m03093)[Fria] Harvey Friedman Metamathematics of comparability Manuscript dated October

2001[Frib] Metamathematics of Ulm theory Manuscript dated November 2001[FSS83] Harvey M Friedman Stephen G Simpson and Rick L Smith Countable algebra and

set existence axioms Ann Pure Appl Logic 25 (1983) no 2 141ndash181 MR725732(85i03157)

[Gao04] Su Gao The homeomorphism problem for countable topological spaces TopologyAppl 139 (2004) no 1-3 97ndash112 MR2051099 (2005e54037)

[GK02] Sergei S Goncharov and Julia Knight Computable structure and antistructure theo-rems Algebra Logika 41 (2002) no 6 639ndash681 757 MR1967769 (2004a03047)

[Hir94] Jeffry L Hirst Reverse mathematics and ordinal exponentiation Ann Pure ApplLogic 66 (1994) no 1 1ndash18 MR1263322 (95a03078)



[Hir00] Reverse mathematics and rank functions for directed graphs Arch MathLogic 39 (2000) no 8 569ndash579 MR1797809 (2001k03134)

[Hir05] Jeffry L Hirst A survey of the reverse mathematics of ordinal arithmetic ReverseMathematics 2001 Lecture Notes in Logic vol 12 Association of Symbolic Logic2005 MR2185437 (2006f03024)

[HK95] Greg Hjorth and Alexander S Kechris Analytic equivalence relations and Ulm-type classifications J Symbolic Logic 60 (1995) no 4 1273ndash1300 MR1367210

(96m54068)[HK01] Recent developments in the theory of Borel reducibility Fund Math 170

(2001) no 1-2 21ndash52 Dedicated to the memory of Jerzy Los MR1881047(2002k03076)

[Kap69] Irving Kaplansky Infinite abelian groups Revised edition The University of MichiganPress Ann Arbor Mich 1969 MR0233887 (382208)

[Kop89] Sabine Koppelberg Handbook of Boolean algebras Vol 1 North-Holland PublishingCo Amsterdam 1989 Edited by J Donald Monk and Robert Bonnet MR991565(90k06002)

[Mon05a] Antonio Montalban Beyond the arithmetic PhD thesis Cornell University IthacaNew York 2005

[Mon05b] Antonio Montalban Up to equimorphism hyperarithmetic is recursive Journal ofSymbolic Logic 70 (2005) no 2 360ndash378 MR2140035 (2006a03063)

[Mun00] James R Munkres Topology second ed Prentice-Hall Inc Englewood Cliffs NJ2000

[Sac90] Gerald E Sacks Higher recursion theory Perspectives in Mathematical LogicSpringer-Verlag Berlin 1990 MR1080970 (92a03062)

[Sel91] V L Selivanov The fine hierarchy and definable index sets Algebra i Logika 30(1991) no 6 705ndash725 771 MR1213731 (95a03057)

[Sho93] Richard A Shore On the strength of Fraıssersquos conjecture Logical methods (IthacaNY 1992) Progr Comput Sci Appl Logic vol 12 Birkhauser Boston Boston MA1993 pp 782ndash813 MR1281145 (95a03002)

[Sho06] Invariants Boolean algebras and ACA+0 Trans Amer Math Soc 358 (2006)

965ndash987 MR2187642 (2006i03013)[Sim99] Stephen G Simpson Subsystems of second order arithmetic Springer 1999

MR1723993 (2001i03126)[Spe55] Clifford Spector Recursive well-orderings J Symb Logic 20 (1955) 151ndash163

MR0074347 (17570b)

Department of Mathematics Notre Dame University Notre Dame Indiana 46556

E-mail address erlkoenigndedu

URL wwwndedusimngreenbeCurrent address School of Mathematics Statistics and Computer Science Victoria University

Wellington New ZealandE-mail address greenbergmcsvuwacnz

URL wwwmcsvuwacnzsimgreenburg

Department of Mathematics University of Chicago Chicago Illinois 60637

E-mail address antoniomathuchicagoedu

URL wwwmathuchicagoedusimantonio


1 Introduction

11 Reverse mathematics

12 The classes

13 The statements

14 Reductions

15 Results

16 More results

2 Ordinals




31 Ranked trees

32 Reductions



41 Definitions

42 Ranked Boolean algebras

43 Reductions


5 Reduced p-groups

51 Ranked p-groups

52 Reductions



61 Definitions

62 Metrizable spaces

63 Compact spaces

64 Ranked spaces

65 Reversals



References


to classify well-founded trees superatomic Boolean algebras and reduced Abelianp-groups In this paper we investigate the proof-theoretic strength of various state-ments related to the existence of invariants and ranks on these classes We doit from the viewpoint of Reverse Mathematics we refer the reader to [Sim99] formore information about the program of Reverse Mathematics We assume that thereader is familiar with at least the introductory chapter of [Sim99]

It turns out that in some sense the structures under consideration effectivelycode the ordinals which are their ranks Thus the study of these structures is closelyrelated to two issues the translation processes between these classes which reducestatements about one class to another (and in particular to ordinals) and thestrength of related questions for the class of ordinals The corresponding subsystemof second-order arithmetic is ATR0 the system which allows us to iterate arithmeticcomprehension along ordinals To quote Simpson ([Sim99 Page 176])

ATR0 is the weakest set of axioms which permits the developmentof a decent theory of countable ordinals

Our general aim is to demonstrate that a similar statement can be made for well-founded trees superatomic Boolean algebras etc general statements about theseclasses will be shown to be equivalent to ATR0 thereby implying the necessary use ofordinal ranks in the investigation of these classes Our work continues investigationsof ordinals (Friedman and Hirst [FH90] see [Hir05] for a survey) of reduced Abelianp-groups (see Simpson [Sim99] and Friedman Simpson and Smith [FSS83]) ofcountable compact metric spaces (see Friedman [Fria] and Friedman and Hirst[FH91]) and of well-founded directed graphs (Hirst [Hir00])

As we mentioned key tools for establishing our results are reductions betweenvarious classes of objects These reductions are an interesting object of studyin their own right Indeed we have two points of view classical in which weinvestigate when there are continuous (or even computable) reductions of one classto another and proof-theoretic in which we ask in what system can one show thatthese reductions indeed preserve notions such as isomorphism and embeddability

11 Reverse mathematics In this paper we only use common subsystems ofsecond-order arithmetic The base theory we use will usually be RCA0 ie thesystem that consists of the semi-ring axioms ∆0

1-comprehension and Σ01-induction

We often use the stronger system ACA0 which adds comprehension for arithmeticformulas We note that over RCA0 ACA0 is equivalent to the existence of the rangeof any one-to-one function f N rarr N We will also occasionally use the systemRCA2 which consists of RCA0 together with Σ0

2-inductionThe focus though is Friedmanrsquos even stronger system ATR0 which enables us to

iterate arithmetical comprehension along any well-ordering As mentioned abovethis is the system which is both sufficient and necessary for a theory of ordinals insecond-order arithmetic For example comparability of well-orderings is equivalentto ATR0 over RCA0

For recursion-theoretic intuition we mention that ATR0 is equivalent to thestatement that for every X sub N and every ordinal α X(α) the αth iterate of theTuring jump of X exists

12 The classes We discuss the various classes of objects with which we deal onlybriefly in this introduction as greater detail will be given at the beginning of eachsection The common feature of these classes is that they form the ldquowell-founded

























(n m) An Am exists

































R-p-G leEW On























2 Ordinals

























γx =sum

yltNxyisinS

(ωβy + 1)





















sim= 17 if ϕ(n)Now








31 Ranked trees


























⋃n 〈n〉Tn









Tn and let rkT






















































































αq + 1 αp






41 Definitions


























andnltσ x





andkltn x

σ(k)k








Now B =⋃





























x It follows that
























of σn 0 or σ











⋂nisinN
















oplusnisinN












sim= Bmexists







n Bn













oplusn Bn Now if




















By Lemma 426 we get





































(g) = 13m1〈g T1〉13m2
































oplusnisinN










t ]Let αn =

sumtisinN






⋃slet βn











5 Reduced p-groups










1-CA0 over RCA0














































decrease

















Similarly





52 Reductions






Also


























sumiltkprime niτi


g(σ) =sum

iltk

mi σi =sum

iltkprime

ni τ primei















61 Definitions



x isin U sube⋂







































63 Compact spaces


such that X sube⋃


that X sube⋃

nisinF Un






lowing
















nk gt n


















nal





⋃llek Unl

























































sumsisinN









sumt βn





























1 This











(xprimei0

) lt rkGr(xi0) Last









g(〈xprime0 x




sumilen mixi = p

sumilen mix



ilen(mip)xi =sum

ilen mixprimei = 0


0 xprimen〉 is



References


































MR0074347 (17570b)










1 Introduction


12 The classes

13 The statements

14 Reductions

15 Results

16 More results

2 Ordinals




31 Ranked trees

32 Reductions



41 Definitions


43 Reductions


5 Reduced p-groups

51 Ranked p-groups

52 Reductions



61 Definitions


63 Compact spaces

64 Ranked spaces

65 Reversals



References
























(n m) An Am exists

































R-p-G leEW On























2 Ordinals

























γx =sum

yltNxyisinS

(ωβy + 1)





















sim= 17 if ϕ(n)Now








31 Ranked trees


























⋃n 〈n〉Tn









Tn and let rkT






















































































αq + 1 αp






41 Definitions


























andnltσ x





andkltn x

σ(k)k








Now B =⋃





























x It follows that
























of σn 0 or σ











⋂nisinN
















oplusnisinN












sim= Bmexists







n Bn













oplusn Bn Now if




















By Lemma 426 we get





































(g) = 13m1〈g T1〉13m2
































oplusnisinN










t ]Let αn =

sumtisinN






⋃slet βn











5 Reduced p-groups










1-CA0 over RCA0














































decrease

















Similarly





52 Reductions






Also


























sumiltkprime niτi


g(σ) =sum

iltk

mi σi =sum

iltkprime

ni τ primei















61 Definitions



x isin U sube⋂







































63 Compact spaces


such that X sube⋃


that X sube⋃

nisinF Un






lowing
















nk gt n


















nal





⋃llek Unl

























































sumsisinN









sumt βn





























1 This











(xprimei0

) lt rkGr(xi0) Last









g(〈xprime0 x




sumilen mixi = p

sumilen mix



ilen(mip)xi =sum

ilen mixprimei = 0


0 xprimen〉 is



References


































MR0074347 (17570b)










1 Introduction


12 The classes

13 The statements

14 Reductions

15 Results

16 More results

2 Ordinals




31 Ranked trees

32 Reductions



41 Definitions


43 Reductions


5 Reduced p-groups

51 Ranked p-groups

52 Reductions



61 Definitions


63 Compact spaces

64 Ranked spaces

65 Reversals



References

























R-p-G leEW On























2 Ordinals

























γx =sum

yltNxyisinS

(ωβy + 1)





















sim= 17 if ϕ(n)Now








31 Ranked trees


























⋃n 〈n〉Tn









Tn and let rkT






















































































αq + 1 αp






41 Definitions


























andnltσ x





andkltn x

σ(k)k








Now B =⋃





























x It follows that
























of σn 0 or σ











⋂nisinN
















oplusnisinN












sim= Bmexists







n Bn













oplusn Bn Now if




















By Lemma 426 we get





































(g) = 13m1〈g T1〉13m2
































oplusnisinN










t ]Let αn =

sumtisinN






⋃slet βn











5 Reduced p-groups










1-CA0 over RCA0














































decrease

















Similarly





52 Reductions






Also


























sumiltkprime niτi


g(σ) =sum

iltk

mi σi =sum

iltkprime

ni τ primei















61 Definitions



x isin U sube⋂







































63 Compact spaces


such that X sube⋃


that X sube⋃

nisinF Un






lowing
















nk gt n


















nal





⋃llek Unl

























































sumsisinN









sumt βn





























1 This











(xprimei0

) lt rkGr(xi0) Last









g(〈xprime0 x




sumilen mixi = p

sumilen mix



ilen(mip)xi =sum

ilen mixprimei = 0


0 xprimen〉 is



References


































MR0074347 (17570b)










1 Introduction


12 The classes

13 The statements

14 Reductions

15 Results

16 More results

2 Ordinals




31 Ranked trees

32 Reductions



41 Definitions


43 Reductions


5 Reduced p-groups

51 Ranked p-groups

52 Reductions



61 Definitions


63 Compact spaces

64 Ranked spaces

65 Reversals



References



















2 Ordinals

























γx =sum

yltNxyisinS

(ωβy + 1)





















sim= 17 if ϕ(n)Now








31 Ranked trees


























⋃n 〈n〉Tn









Tn and let rkT






















































































αq + 1 αp






41 Definitions


























andnltσ x





andkltn x

σ(k)k








Now B =⋃





























x It follows that
























of σn 0 or σ











⋂nisinN
















oplusnisinN












sim= Bmexists







n Bn













oplusn Bn Now if




















By Lemma 426 we get





































(g) = 13m1〈g T1〉13m2
































oplusnisinN










t ]Let αn =

sumtisinN






⋃slet βn











5 Reduced p-groups










1-CA0 over RCA0














































decrease

















Similarly





52 Reductions






Also


























sumiltkprime niτi


g(σ) =sum

iltk

mi σi =sum

iltkprime

ni τ primei















61 Definitions



x isin U sube⋂







































63 Compact spaces


such that X sube⋃


that X sube⋃

nisinF Un






lowing
















nk gt n


















nal





⋃llek Unl

























































sumsisinN









sumt βn





























1 This











(xprimei0

) lt rkGr(xi0) Last









g(〈xprime0 x




sumilen mixi = p

sumilen mix



ilen(mip)xi =sum

ilen mixprimei = 0


0 xprimen〉 is



References


































MR0074347 (17570b)










1 Introduction


12 The classes

13 The statements

14 Reductions

15 Results

16 More results

2 Ordinals




31 Ranked trees

32 Reductions



41 Definitions


43 Reductions


5 Reduced p-groups

51 Ranked p-groups

52 Reductions



61 Definitions


63 Compact spaces

64 Ranked spaces

65 Reversals



References






















γx =sum

yltNxyisinS

(ωβy + 1)





















sim= 17 if ϕ(n)Now








31 Ranked trees


























⋃n 〈n〉Tn









Tn and let rkT






















































































αq + 1 αp






41 Definitions


























andnltσ x





andkltn x

σ(k)k








Now B =⋃





























x It follows that
























of σn 0 or σ











⋂nisinN
















oplusnisinN












sim= Bmexists







n Bn













oplusn Bn Now if




















By Lemma 426 we get





































(g) = 13m1〈g T1〉13m2
































oplusnisinN










t ]Let αn =

sumtisinN






⋃slet βn











5 Reduced p-groups










1-CA0 over RCA0














































decrease

















Similarly





52 Reductions






Also


























sumiltkprime niτi


g(σ) =sum

iltk

mi σi =sum

iltkprime

ni τ primei















61 Definitions



x isin U sube⋂







































63 Compact spaces


such that X sube⋃


that X sube⋃

nisinF Un






lowing
















nk gt n


















nal





⋃llek Unl

























































sumsisinN









sumt βn





























1 This











(xprimei0

) lt rkGr(xi0) Last









g(〈xprime0 x




sumilen mixi = p

sumilen mix



ilen(mip)xi =sum

ilen mixprimei = 0


0 xprimen〉 is



References


































MR0074347 (17570b)










1 Introduction


12 The classes

13 The statements

14 Reductions

15 Results

16 More results

2 Ordinals




31 Ranked trees

32 Reductions



41 Definitions


43 Reductions


5 Reduced p-groups

51 Ranked p-groups

52 Reductions



61 Definitions


63 Compact spaces

64 Ranked spaces

65 Reversals



References




sim= 17 if ϕ(n)Now








31 Ranked trees


























⋃n 〈n〉Tn









Tn and let rkT






















































































αq + 1 αp






41 Definitions


























andnltσ x





andkltn x

σ(k)k








Now B =⋃





























x It follows that
























of σn 0 or σ











⋂nisinN
















oplusnisinN












sim= Bmexists







n Bn













oplusn Bn Now if




















By Lemma 426 we get





































(g) = 13m1〈g T1〉13m2
































oplusnisinN










t ]Let αn =

sumtisinN






⋃slet βn











5 Reduced p-groups










1-CA0 over RCA0














































decrease

















Similarly





52 Reductions






Also


























sumiltkprime niτi


g(σ) =sum

iltk

mi σi =sum

iltkprime

ni τ primei















61 Definitions



x isin U sube⋂







































63 Compact spaces


such that X sube⋃


that X sube⋃

nisinF Un






lowing
















nk gt n


















nal





⋃llek Unl

























































sumsisinN









sumt βn





























1 This











(xprimei0

) lt rkGr(xi0) Last









g(〈xprime0 x




sumilen mixi = p

sumilen mix



ilen(mip)xi =sum

ilen mixprimei = 0


0 xprimen〉 is



References


































MR0074347 (17570b)










1 Introduction


12 The classes

13 The statements

14 Reductions

15 Results

16 More results

2 Ordinals




31 Ranked trees

32 Reductions



41 Definitions


43 Reductions


5 Reduced p-groups

51 Ranked p-groups

52 Reductions



61 Definitions


63 Compact spaces

64 Ranked spaces

65 Reversals



References















⋃n 〈n〉Tn









Tn and let rkT






















































































αq + 1 αp






41 Definitions


























andnltσ x





andkltn x

σ(k)k








Now B =⋃





























x It follows that
























of σn 0 or σ











⋂nisinN
















oplusnisinN












sim= Bmexists







n Bn













oplusn Bn Now if




















By Lemma 426 we get





































(g) = 13m1〈g T1〉13m2
































oplusnisinN










t ]Let αn =

sumtisinN






⋃slet βn











5 Reduced p-groups










1-CA0 over RCA0














































decrease

















Similarly





52 Reductions






Also


























sumiltkprime niτi


g(σ) =sum

iltk

mi σi =sum

iltkprime

ni τ primei















61 Definitions



x isin U sube⋂







































63 Compact spaces


such that X sube⋃


that X sube⋃

nisinF Un






lowing
















nk gt n


















nal





⋃llek Unl

























































sumsisinN









sumt βn





























1 This











(xprimei0

) lt rkGr(xi0) Last









g(〈xprime0 x




sumilen mixi = p

sumilen mix



ilen(mip)xi =sum

ilen mixprimei = 0


0 xprimen〉 is



References


































MR0074347 (17570b)










1 Introduction


12 The classes

13 The statements

14 Reductions

15 Results

16 More results

2 Ordinals




31 Ranked trees

32 Reductions



41 Definitions


43 Reductions


5 Reduced p-groups

51 Ranked p-groups

52 Reductions



61 Definitions


63 Compact spaces

64 Ranked spaces

65 Reversals



References




Tn and let rkT






















































































αq + 1 αp






41 Definitions


























andnltσ x





andkltn x

σ(k)k








Now B =⋃





























x It follows that
























of σn 0 or σ











⋂nisinN
















oplusnisinN












sim= Bmexists







n Bn













oplusn Bn Now if




















By Lemma 426 we get





































(g) = 13m1〈g T1〉13m2
































oplusnisinN










t ]Let αn =

sumtisinN






⋃slet βn











5 Reduced p-groups










1-CA0 over RCA0














































decrease

















Similarly





52 Reductions






Also


























sumiltkprime niτi


g(σ) =sum

iltk

mi σi =sum

iltkprime

ni τ primei















61 Definitions



x isin U sube⋂







































63 Compact spaces


such that X sube⋃


that X sube⋃

nisinF Un






lowing
















nk gt n


















nal





⋃llek Unl

























































sumsisinN









sumt βn





























1 This











(xprimei0

) lt rkGr(xi0) Last









g(〈xprime0 x




sumilen mixi = p

sumilen mix



ilen(mip)xi =sum

ilen mixprimei = 0


0 xprimen〉 is



References


































MR0074347 (17570b)










1 Introduction


12 The classes

13 The statements

14 Reductions

15 Results

16 More results

2 Ordinals




31 Ranked trees

32 Reductions



41 Definitions


43 Reductions


5 Reduced p-groups

51 Ranked p-groups

52 Reductions



61 Definitions


63 Compact spaces

64 Ranked spaces

65 Reversals



References





































































αq + 1 αp






41 Definitions


























andnltσ x





andkltn x

σ(k)k








Now B =⋃





























x It follows that
























of σn 0 or σ











⋂nisinN
















oplusnisinN












sim= Bmexists







n Bn













oplusn Bn Now if




















By Lemma 426 we get





































(g) = 13m1〈g T1〉13m2
































oplusnisinN










t ]Let αn =

sumtisinN






⋃slet βn











5 Reduced p-groups










1-CA0 over RCA0














































decrease

















Similarly





52 Reductions






Also


























sumiltkprime niτi


g(σ) =sum

iltk

mi σi =sum

iltkprime

ni τ primei















61 Definitions



x isin U sube⋂







































63 Compact spaces


such that X sube⋃


that X sube⋃

nisinF Un






lowing
















nk gt n


















nal





⋃llek Unl

























































sumsisinN









sumt βn





























1 This











(xprimei0

) lt rkGr(xi0) Last









g(〈xprime0 x




sumilen mixi = p

sumilen mix



ilen(mip)xi =sum

ilen mixprimei = 0


0 xprimen〉 is



References


































MR0074347 (17570b)










1 Introduction


12 The classes

13 The statements

14 Reductions

15 Results

16 More results

2 Ordinals




31 Ranked trees

32 Reductions



41 Definitions


43 Reductions


5 Reduced p-groups

51 Ranked p-groups

52 Reductions



61 Definitions


63 Compact spaces

64 Ranked spaces

65 Reversals



References






















andnltσ x





andkltn x

σ(k)k








Now B =⋃





























x It follows that
























of σn 0 or σ











⋂nisinN
















oplusnisinN












sim= Bmexists







n Bn













oplusn Bn Now if




















By Lemma 426 we get





































(g) = 13m1〈g T1〉13m2
































oplusnisinN










t ]Let αn =

sumtisinN






⋃slet βn











5 Reduced p-groups










1-CA0 over RCA0














































decrease

















Similarly





52 Reductions






Also


























sumiltkprime niτi


g(σ) =sum

iltk

mi σi =sum

iltkprime

ni τ primei















61 Definitions



x isin U sube⋂







































63 Compact spaces


such that X sube⋃


that X sube⋃

nisinF Un






lowing
















nk gt n


















nal





⋃llek Unl

























































sumsisinN









sumt βn





























1 This











(xprimei0

) lt rkGr(xi0) Last









g(〈xprime0 x




sumilen mixi = p

sumilen mix



ilen(mip)xi =sum

ilen mixprimei = 0


0 xprimen〉 is



References


































MR0074347 (17570b)










1 Introduction


12 The classes

13 The statements

14 Reductions

15 Results

16 More results

2 Ordinals




31 Ranked trees

32 Reductions



41 Definitions


43 Reductions


5 Reduced p-groups

51 Ranked p-groups

52 Reductions



61 Definitions


63 Compact spaces

64 Ranked spaces

65 Reversals



References




















x It follows that
























of σn 0 or σ











⋂nisinN
















oplusnisinN












sim= Bmexists







n Bn













oplusn Bn Now if




















By Lemma 426 we get





































(g) = 13m1〈g T1〉13m2
































oplusnisinN










t ]Let αn =

sumtisinN






⋃slet βn











5 Reduced p-groups










1-CA0 over RCA0














































decrease

















Similarly





52 Reductions






Also


























sumiltkprime niτi


g(σ) =sum

iltk

mi σi =sum

iltkprime

ni τ primei















61 Definitions



x isin U sube⋂







































63 Compact spaces


such that X sube⋃


that X sube⋃

nisinF Un






lowing
















nk gt n


















nal





⋃llek Unl

























































sumsisinN









sumt βn





























1 This











(xprimei0

) lt rkGr(xi0) Last









g(〈xprime0 x




sumilen mixi = p

sumilen mix



ilen(mip)xi =sum

ilen mixprimei = 0


0 xprimen〉 is



References


































MR0074347 (17570b)










1 Introduction


12 The classes

13 The statements

14 Reductions

15 Results

16 More results

2 Ordinals




31 Ranked trees

32 Reductions



41 Definitions


43 Reductions


5 Reduced p-groups

51 Ranked p-groups

52 Reductions



61 Definitions


63 Compact spaces

64 Ranked spaces

65 Reversals



References























of σn 0 or σ











⋂nisinN
















oplusnisinN












sim= Bmexists







n Bn













oplusn Bn Now if




















By Lemma 426 we get





































(g) = 13m1〈g T1〉13m2
































oplusnisinN










t ]Let αn =

sumtisinN






⋃slet βn











5 Reduced p-groups










1-CA0 over RCA0














































decrease

















Similarly





52 Reductions






Also


























sumiltkprime niτi


g(σ) =sum

iltk

mi σi =sum

iltkprime

ni τ primei















61 Definitions



x isin U sube⋂







































63 Compact spaces


such that X sube⋃


that X sube⋃

nisinF Un






lowing
















nk gt n


















nal





⋃llek Unl

























































sumsisinN









sumt βn





























1 This











(xprimei0

) lt rkGr(xi0) Last









g(〈xprime0 x




sumilen mixi = p

sumilen mix



ilen(mip)xi =sum

ilen mixprimei = 0


0 xprimen〉 is



References


































MR0074347 (17570b)










1 Introduction


12 The classes

13 The statements

14 Reductions

15 Results

16 More results

2 Ordinals




31 Ranked trees

32 Reductions



41 Definitions


43 Reductions


5 Reduced p-groups

51 Ranked p-groups

52 Reductions



61 Definitions


63 Compact spaces

64 Ranked spaces

65 Reversals



References





⋂nisinN
















oplusnisinN












sim= Bmexists







n Bn













oplusn Bn Now if




















By Lemma 426 we get





































(g) = 13m1〈g T1〉13m2
































oplusnisinN










t ]Let αn =

sumtisinN






⋃slet βn











5 Reduced p-groups










1-CA0 over RCA0














































decrease

















Similarly





52 Reductions






Also


























sumiltkprime niτi


g(σ) =sum

iltk

mi σi =sum

iltkprime

ni τ primei















61 Definitions



x isin U sube⋂







































63 Compact spaces


such that X sube⋃


that X sube⋃

nisinF Un






lowing
















nk gt n


















nal





⋃llek Unl

























































sumsisinN









sumt βn





























1 This











(xprimei0

) lt rkGr(xi0) Last









g(〈xprime0 x




sumilen mixi = p

sumilen mix



ilen(mip)xi =sum

ilen mixprimei = 0


0 xprimen〉 is



References


































MR0074347 (17570b)










1 Introduction


12 The classes

13 The statements

14 Reductions

15 Results

16 More results

2 Ordinals




31 Ranked trees

32 Reductions



41 Definitions


43 Reductions


5 Reduced p-groups

51 Ranked p-groups

52 Reductions



61 Definitions


63 Compact spaces

64 Ranked spaces

65 Reversals



References








sim= Bmexists







n Bn













oplusn Bn Now if




















By Lemma 426 we get





































(g) = 13m1〈g T1〉13m2
































oplusnisinN










t ]Let αn =

sumtisinN






⋃slet βn











5 Reduced p-groups










1-CA0 over RCA0














































decrease

















Similarly





52 Reductions






Also


























sumiltkprime niτi


g(σ) =sum

iltk

mi σi =sum

iltkprime

ni τ primei















61 Definitions



x isin U sube⋂







































63 Compact spaces


such that X sube⋃


that X sube⋃

nisinF Un






lowing
















nk gt n


















nal





⋃llek Unl

























































sumsisinN









sumt βn





























1 This











(xprimei0

) lt rkGr(xi0) Last









g(〈xprime0 x




sumilen mixi = p

sumilen mix



ilen(mip)xi =sum

ilen mixprimei = 0


0 xprimen〉 is



References


































MR0074347 (17570b)










1 Introduction


12 The classes

13 The statements

14 Reductions

15 Results

16 More results

2 Ordinals




31 Ranked trees

32 Reductions



41 Definitions


43 Reductions


5 Reduced p-groups

51 Ranked p-groups

52 Reductions



61 Definitions


63 Compact spaces

64 Ranked spaces

65 Reversals



References





oplusn Bn Now if




















By Lemma 426 we get





































(g) = 13m1〈g T1〉13m2
































oplusnisinN










t ]Let αn =

sumtisinN






⋃slet βn











5 Reduced p-groups










1-CA0 over RCA0














































decrease

















Similarly





52 Reductions






Also


























sumiltkprime niτi


g(σ) =sum

iltk

mi σi =sum

iltkprime

ni τ primei















61 Definitions



x isin U sube⋂







































63 Compact spaces


such that X sube⋃


that X sube⋃

nisinF Un






lowing
















nk gt n


















nal





⋃llek Unl

























































sumsisinN









sumt βn





























1 This











(xprimei0

) lt rkGr(xi0) Last









g(〈xprime0 x




sumilen mixi = p

sumilen mix



ilen(mip)xi =sum

ilen mixprimei = 0


0 xprimen〉 is



References


































MR0074347 (17570b)










1 Introduction


12 The classes

13 The statements

14 Reductions

15 Results

16 More results

2 Ordinals




31 Ranked trees

32 Reductions



41 Definitions


43 Reductions


5 Reduced p-groups

51 Ranked p-groups

52 Reductions



61 Definitions


63 Compact spaces

64 Ranked spaces

65 Reversals



References










By Lemma 426 we get





































(g) = 13m1〈g T1〉13m2
































oplusnisinN










t ]Let αn =

sumtisinN






⋃slet βn











5 Reduced p-groups










1-CA0 over RCA0














































decrease

















Similarly





52 Reductions






Also


























sumiltkprime niτi


g(σ) =sum

iltk

mi σi =sum

iltkprime

ni τ primei















61 Definitions



x isin U sube⋂







































63 Compact spaces


such that X sube⋃


that X sube⋃

nisinF Un






lowing
















nk gt n


















nal





⋃llek Unl

























































sumsisinN









sumt βn





























1 This











(xprimei0

) lt rkGr(xi0) Last









g(〈xprime0 x




sumilen mixi = p

sumilen mix



ilen(mip)xi =sum

ilen mixprimei = 0


0 xprimen〉 is



References


































MR0074347 (17570b)










1 Introduction


12 The classes

13 The statements

14 Reductions

15 Results

16 More results

2 Ordinals




31 Ranked trees

32 Reductions



41 Definitions


43 Reductions


5 Reduced p-groups

51 Ranked p-groups

52 Reductions



61 Definitions


63 Compact spaces

64 Ranked spaces

65 Reversals



References























(g) = 13m1〈g T1〉13m2
































oplusnisinN










t ]Let αn =

sumtisinN






⋃slet βn











5 Reduced p-groups










1-CA0 over RCA0














































decrease

















Similarly





52 Reductions






Also


























sumiltkprime niτi


g(σ) =sum

iltk

mi σi =sum

iltkprime

ni τ primei















61 Definitions



x isin U sube⋂







































63 Compact spaces


such that X sube⋃


that X sube⋃

nisinF Un






lowing
















nk gt n


















nal





⋃llek Unl

























































sumsisinN









sumt βn





























1 This











(xprimei0

) lt rkGr(xi0) Last









g(〈xprime0 x




sumilen mixi = p

sumilen mix



ilen(mip)xi =sum

ilen mixprimei = 0


0 xprimen〉 is



References


































MR0074347 (17570b)










1 Introduction


12 The classes

13 The statements

14 Reductions

15 Results

16 More results

2 Ordinals




31 Ranked trees

32 Reductions



41 Definitions


43 Reductions


5 Reduced p-groups

51 Ranked p-groups

52 Reductions



61 Definitions


63 Compact spaces

64 Ranked spaces

65 Reversals



References


























oplusnisinN










t ]Let αn =

sumtisinN






⋃slet βn











5 Reduced p-groups










1-CA0 over RCA0














































decrease

















Similarly





52 Reductions






Also


























sumiltkprime niτi


g(σ) =sum

iltk

mi σi =sum

iltkprime

ni τ primei















61 Definitions



x isin U sube⋂







































63 Compact spaces


such that X sube⋃


that X sube⋃

nisinF Un






lowing
















nk gt n


















nal





⋃llek Unl

























































sumsisinN









sumt βn





























1 This











(xprimei0

) lt rkGr(xi0) Last









g(〈xprime0 x




sumilen mixi = p

sumilen mix



ilen(mip)xi =sum

ilen mixprimei = 0


0 xprimen〉 is



References


































MR0074347 (17570b)










1 Introduction


12 The classes

13 The statements

14 Reductions

15 Results

16 More results

2 Ordinals




31 Ranked trees

32 Reductions



41 Definitions


43 Reductions


5 Reduced p-groups

51 Ranked p-groups

52 Reductions



61 Definitions


63 Compact spaces

64 Ranked spaces

65 Reversals



References





5 Reduced p-groups










1-CA0 over RCA0














































decrease

















Similarly





52 Reductions






Also


























sumiltkprime niτi


g(σ) =sum

iltk

mi σi =sum

iltkprime

ni τ primei















61 Definitions



x isin U sube⋂







































63 Compact spaces


such that X sube⋃


that X sube⋃

nisinF Un






lowing
















nk gt n


















nal





⋃llek Unl

























































sumsisinN









sumt βn





























1 This











(xprimei0

) lt rkGr(xi0) Last









g(〈xprime0 x




sumilen mixi = p

sumilen mix



ilen(mip)xi =sum

ilen mixprimei = 0


0 xprimen〉 is



References


































MR0074347 (17570b)










1 Introduction


12 The classes

13 The statements

14 Reductions

15 Results

16 More results

2 Ordinals




31 Ranked trees

32 Reductions



41 Definitions


43 Reductions


5 Reduced p-groups

51 Ranked p-groups

52 Reductions



61 Definitions


63 Compact spaces

64 Ranked spaces

65 Reversals



References







































decrease

















Similarly





52 Reductions






Also


























sumiltkprime niτi


g(σ) =sum

iltk

mi σi =sum

iltkprime

ni τ primei















61 Definitions



x isin U sube⋂







































63 Compact spaces


such that X sube⋃


that X sube⋃

nisinF Un






lowing
















nk gt n


















nal





⋃llek Unl

























































sumsisinN









sumt βn





























1 This











(xprimei0

) lt rkGr(xi0) Last









g(〈xprime0 x




sumilen mixi = p

sumilen mix



ilen(mip)xi =sum

ilen mixprimei = 0


0 xprimen〉 is



References


































MR0074347 (17570b)










1 Introduction


12 The classes

13 The statements

14 Reductions

15 Results

16 More results

2 Ordinals




31 Ranked trees

32 Reductions



41 Definitions


43 Reductions


5 Reduced p-groups

51 Ranked p-groups

52 Reductions



61 Definitions


63 Compact spaces

64 Ranked spaces

65 Reversals



References


52 Reductions






Also


























sumiltkprime niτi


g(σ) =sum

iltk

mi σi =sum

iltkprime

ni τ primei















61 Definitions



x isin U sube⋂







































63 Compact spaces


such that X sube⋃


that X sube⋃

nisinF Un






lowing
















nk gt n


















nal





⋃llek Unl

























































sumsisinN









sumt βn





























1 This











(xprimei0

) lt rkGr(xi0) Last









g(〈xprime0 x




sumilen mixi = p

sumilen mix



ilen(mip)xi =sum

ilen mixprimei = 0


0 xprimen〉 is



References


































MR0074347 (17570b)










1 Introduction


12 The classes

13 The statements

14 Reductions

15 Results

16 More results

2 Ordinals




31 Ranked trees

32 Reductions



41 Definitions


43 Reductions


5 Reduced p-groups

51 Ranked p-groups

52 Reductions



61 Definitions


63 Compact spaces

64 Ranked spaces

65 Reversals



References












sumiltkprime niτi


g(σ) =sum

iltk

mi σi =sum

iltkprime

ni τ primei















61 Definitions



x isin U sube⋂







































63 Compact spaces


such that X sube⋃


that X sube⋃

nisinF Un






lowing
















nk gt n


















nal





⋃llek Unl

























































sumsisinN









sumt βn





























1 This











(xprimei0

) lt rkGr(xi0) Last









g(〈xprime0 x




sumilen mixi = p

sumilen mix



ilen(mip)xi =sum

ilen mixprimei = 0


0 xprimen〉 is



References


































MR0074347 (17570b)










1 Introduction


12 The classes

13 The statements

14 Reductions

15 Results

16 More results

2 Ordinals




31 Ranked trees

32 Reductions



41 Definitions


43 Reductions


5 Reduced p-groups

51 Ranked p-groups

52 Reductions



61 Definitions


63 Compact spaces

64 Ranked spaces

65 Reversals



References




61 Definitions



x isin U sube⋂







































63 Compact spaces


such that X sube⋃


that X sube⋃

nisinF Un






lowing
















nk gt n


















nal





⋃llek Unl

























































sumsisinN









sumt βn





























1 This











(xprimei0

) lt rkGr(xi0) Last









g(〈xprime0 x




sumilen mixi = p

sumilen mix



ilen(mip)xi =sum

ilen mixprimei = 0


0 xprimen〉 is



References


































MR0074347 (17570b)










1 Introduction


12 The classes

13 The statements

14 Reductions

15 Results

16 More results

2 Ordinals




31 Ranked trees

32 Reductions



41 Definitions


43 Reductions


5 Reduced p-groups

51 Ranked p-groups

52 Reductions



61 Definitions


63 Compact spaces

64 Ranked spaces

65 Reversals



References



























63 Compact spaces


such that X sube⋃


that X sube⋃

nisinF Un






lowing
















nk gt n


















nal





⋃llek Unl

























































sumsisinN









sumt βn





























1 This











(xprimei0

) lt rkGr(xi0) Last









g(〈xprime0 x




sumilen mixi = p

sumilen mix



ilen(mip)xi =sum

ilen mixprimei = 0


0 xprimen〉 is



References


































MR0074347 (17570b)










1 Introduction


12 The classes

13 The statements

14 Reductions

15 Results

16 More results

2 Ordinals




31 Ranked trees

32 Reductions



41 Definitions


43 Reductions


5 Reduced p-groups

51 Ranked p-groups

52 Reductions



61 Definitions


63 Compact spaces

64 Ranked spaces

65 Reversals



References



lowing
















nk gt n


















nal





⋃llek Unl

























































sumsisinN









sumt βn





























1 This











(xprimei0

) lt rkGr(xi0) Last









g(〈xprime0 x




sumilen mixi = p

sumilen mix



ilen(mip)xi =sum

ilen mixprimei = 0


0 xprimen〉 is



References


































MR0074347 (17570b)










1 Introduction


12 The classes

13 The statements

14 Reductions

15 Results

16 More results

2 Ordinals




31 Ranked trees

32 Reductions



41 Definitions


43 Reductions


5 Reduced p-groups

51 Ranked p-groups

52 Reductions



61 Definitions


63 Compact spaces

64 Ranked spaces

65 Reversals



References












nal





⋃llek Unl

























































sumsisinN









sumt βn





























1 This











(xprimei0

) lt rkGr(xi0) Last









g(〈xprime0 x




sumilen mixi = p

sumilen mix



ilen(mip)xi =sum

ilen mixprimei = 0


0 xprimen〉 is



References


































MR0074347 (17570b)










1 Introduction


12 The classes

13 The statements

14 Reductions

15 Results

16 More results

2 Ordinals




31 Ranked trees

32 Reductions



41 Definitions


43 Reductions


5 Reduced p-groups

51 Ranked p-groups

52 Reductions



61 Definitions


63 Compact spaces

64 Ranked spaces

65 Reversals



References

















































sumsisinN









sumt βn





























1 This











(xprimei0

) lt rkGr(xi0) Last









g(〈xprime0 x




sumilen mixi = p

sumilen mix



ilen(mip)xi =sum

ilen mixprimei = 0


0 xprimen〉 is



References


































MR0074347 (17570b)










1 Introduction


12 The classes

13 The statements

14 Reductions

15 Results

16 More results

2 Ordinals




31 Ranked trees

32 Reductions



41 Definitions


43 Reductions


5 Reduced p-groups

51 Ranked p-groups

52 Reductions



61 Definitions


63 Compact spaces

64 Ranked spaces

65 Reversals



References























1 This











(xprimei0

) lt rkGr(xi0) Last









g(〈xprime0 x




sumilen mixi = p

sumilen mix



ilen(mip)xi =sum

ilen mixprimei = 0


0 xprimen〉 is



References


































MR0074347 (17570b)










1 Introduction


12 The classes

13 The statements

14 Reductions

15 Results

16 More results

2 Ordinals




31 Ranked trees

32 Reductions



41 Definitions


43 Reductions


5 Reduced p-groups

51 Ranked p-groups

52 Reductions



61 Definitions


63 Compact spaces

64 Ranked spaces

65 Reversals



References










(xprimei0

) lt rkGr(xi0) Last









g(〈xprime0 x




sumilen mixi = p

sumilen mix



ilen(mip)xi =sum

ilen mixprimei = 0


0 xprimen〉 is



References


































MR0074347 (17570b)










1 Introduction


12 The classes

13 The statements

14 Reductions

15 Results

16 More results

2 Ordinals




31 Ranked trees

32 Reductions



41 Definitions


43 Reductions


5 Reduced p-groups

51 Ranked p-groups

52 Reductions



61 Definitions


63 Compact spaces

64 Ranked spaces

65 Reversals



References


















MR0074347 (17570b)










1 Introduction


12 The classes

13 The statements

14 Reductions

15 Results

16 More results

2 Ordinals




31 Ranked trees

32 Reductions



41 Definitions


43 Reductions


5 Reduced p-groups

51 Ranked p-groups

52 Reductions



61 Definitions


63 Compact spaces

64 Ranked spaces

65 Reversals



References

introduction · 1.1. reverse mathematics. in this paper we only use common subsystems of...

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