mirna dzamonja and saharon shelah- on the existence of universal models

8/3/2019 Mirna Dzamonja and Saharon Shelah- On the existence of universal models

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On the existence of universal models

Mirna Dzamonja

School of Mathematics

University of East Anglia

Norwich, NR4 7TJ, UK

[email protected]

http://www.mth.uea.ac.uk/people/md.html

Saharon Shelah

Mathematics Department

Hebrew University of Jerusalem

91904 Givat Ram, Israel

[email protected]

http://shelah.logic.at/

Abstract

Suppose that = ++. In fact, we workmore generally with abstract elementary classes. The criterion forthe consistent existence of universals applies to various well knowntheories, such as triangle-free graphs and simple theories.

Having in mind possible applications in analysis, we further ob-serve that for such , for any fixed > + regular with =

+

, itis consistent that 2 = and there is no normed vector space overQ of size < which is universal for normed vector spaces over Q ofdimension + under the notion of embedding h which specifies (a, b)such that ||h(x)||/||x|| (a, b) for all x.

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1

0 Introduction.

We study the existence of universal models for certain natural theories, which

are not necessarily first order. This paper is self-contained, and it continues

Saharon Shelahs [Sh 457] and [Sh 500]. An example of a theory to which

our results can be applied is the theory of triangle-free graphs, or any simple

theory (in the sense of [Sh 93]). For T a theory with a fixed notion of an

embedding between its models, we say that a model M

ofT is universal formodels of T (of size ) if every model M of T of size , embeds into M.

We similarly define when a family of models is jointly universal for models

of size . More generally, we consider universals in an abstract elementary

class, see Definition 1.9.

Two well known theorems on the existence of universal models for first

order theories T (see [ChKe]) are

1. Under GCH, there is a universal model of T of cardinality for every

> |T|.

2. If 2 |T|, then there is a universal model of T of cardinality .

Without the above assumptions, it tends to be hard for a first order theory to

have a universal model, see [Sh 457] for a discussion and further references.

Although the problem of the existence of universal models for first-order

theories (i.e. elementary classes of all models of such a theory) is the one

which has been studied most extensively, there are of course many natural

theories which are not first order. To approach such questions, we view

the problem from the point of view of abstract elementary classes, which

were introduced in [Sh 88] (in 1 we recall the definitions), and in a more1In the list of publications of S. Shelah, this is publication number 614. Both authors

thank the United States-Israel Binational Science Foundation for a partial support and

various readers of the manuscript. for their helpful comments.

AMS Subject Classification: 03E35, 03C55.

Keywords: universal models, approximation families, consistency results.

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specialized form earlier by Bjarni Jonsson, see [ChKe]. Such classes will bethroughout denoted by K, and if is a cardinal, the family of elements of K

which have size will be denoted by K.

In [Sh 457] S. Shelah introduced the notion of an approximation family

and studied abstract elementary classes with a simple (here called work-

able, to differentiate them from simple theories in the sense of [Sh 93])

-approximation family. One of the results mentioned in [Sh 457] is that for

an uncountable cardinal satisfying =


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As we intend to iterate with (< )-supports, our chain condition has tobe a strong version of +-cc, so to be preserved under such iterations. The

one we use is from [Sh 288], which is also the one used in [Sh 457]. This

condition is a weakening of stationary +-cc. We recall the definition at the

beginning of 2. The question now becomes which -approximation families

yield such a chain condition. We call such approximation families workable.

This notion is defined in 1. In [Sh 457] it is shown that triangle free graphs

and the theory of an indexed family of independent equivalence equations

have workable -approximation families.

In [Sh 500] and elsewhere, S. Shelah expresses the view that the existence

of universal models has relevance to the general problem of classifying un-

stable theories. With this in mind, we can consider a theory as simple if it

has a workable approximation family. In [Sh 93], another meaning of sim-

plicity is considered: a theory is called simple if it does not have the tree

property. In [Sh 500] it was shown that complete simple first order theories

of size < have workable approximation families in . This can be un-

derstood as showing that all simple theories behave better in the respect

of universality than the linear orders do, as it is known by [KjSh 409] that

when GCH fails, linear orders can have a universal in only a few cardi-

nals. The hope of finding dividing lines via the existence of universal modelsis also realized for some non-simple theories, as it was shown by S. She-

lah in [Sh 457] that some non-simple theories have workable approximation

families, like the triangle-free graphs and the theory of an indexed family

of independent equivalence relations, as simplest prototypes of non-simple

theories. In [Sh 500], S. Shelah introduced a hierarchy NSOP n for 3 n

with the intention of encapturing by a formal notion the class of first order

theories which behave nicely with respect to having universal models. Our

research here continues [Sh 457].

We now give an idea of the proof of the positive consistency results.Details are explained in 1 and 2. The idea is that through a (< )-supports

iteration of (< )-complete forcing we obtain the situation under which to

every workable strong -approximation family Kap there corresponds a tree

of elements of Kap. If Kap approximates K and K is nice enough, then the

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models in this tree are organized so that the entire tree can be amalgamatedto a model in K+. Along the iteration we also make sure that every element

of K+ can be embedded into a model obtained as the union of one branch

of such a tree. There are ++ trees used for every approximation family, so

the universal model obtained has size ++. Every individual forcing used in

the iteration has , but the proof of this for > 0 requires us to introduce

an auxiliary step in the forcing.

In 3 we give a consistency result showing that with the same assumptions

on + as above it is consistent that there is no universal normed vector space

of size +, even under a rather weak notion of embedding. We note that

negative consistencyresults relevant to the universality problem tend to be

much easier to obtain than the positive ones, especially as far as the first

order theories are concerned.

We finish this introduction by giving more remarks on related results,

and some conventions used throughout the paper.

The pcf theory of S. Shelah has proved to be a useful line of approach to

the negative aspect of the problem of universality. This approach has been

extensively applied by Menachem Kojman and S. Shelah (e.g. to linear orders

[KjSh 409]), and later by each of them separately (M. Kojman on graphs [Kj],

S. Shelah on Abelian groups [Sh 552] e.g). See [Sh 552] for the history andmore references. One of the ideas involved is to use the existence of a club

guessing sequence to prove that no universals exist. A related result of Mirna

Dzamonja in [Dz1] deals with uniform Eberlein compacta, and in [Dz2] she

shows how the universality axioms presented in this paper can be applied

to that class. Among the positive universality results, let us quote a paper

by Rami Grossberg and S. Shelah [GrSh 174], in which it is shown that e.g.

the class of locally finite groups has a universal model in any strong limit of

cofinality 0 above a compact cardinal. This paper is also the first reference

to the consideration of the universality spectrum as a useful dividing line inmodel theory.

Further positive consistency results appear e.g in S. Shelahs [Sh 100]

where the consistent existence of a universal linear order at 1 with the

negation of CH is shown, and in S. Shelahs [Sh 175], [Sh 175a] where the

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consistency of the existence of a universal graph at for which there is satisfying =


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(5) For a model M, we use |M| to denote the underlying set ofM, and hence||M|| to denote the cardinality of |M|.

1 Approximation families.

Definition 1.1. [Sh 457] Given an infinite cardinal, and u1, u2 +.

A function h : u1 u2 is said to be lawful iff it is 1-1 and for all u1we have h() + = + .

Notation 1.2. (1) For A +, let

(A)def= min{ : A & |}.

If M is a model, we let (M)def= (|M|).

(2) In the following, we shall use the notation M for M (M) (the

meaning of and M will be described in the following definition).

(3) Evdef= {2 : < +}.

Remark 1.3. The notion of divisibility of ordinals used here is that |

means that = [not = ] for some . The intuition behind the

definition of a lawful function is that one regards + as partitioned into blocks

of length , and then a function is lawful iff it acts by permuting within each

block. Then the function (A) simply measures how far the blocks go that

meet A.

Definition 1.4. [Sh 457] Let be an infinite cardinal.

(1) Pair Kap = (Kap, Kap) is a weak -approximation family iff for some

(not necessarily strictly) increasing sequence2

= i : i < + & |i

of finitary vocabularies, each of size we have

2For the applications mentioned in this paper, in the following definitions readers can

restrict their attention to the situation of i = 0 for all i.

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(a) Kap is a set partially ordered by Kap , and such that

M Kap = M is a (M)-model.

(b) If M Kap, then |M| [+]


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() if M Kap and h is a lawful mapping from |M| onto someu +, then for some M Kap we have that |M| = u and

h is a lawful |M|-isomorphism from M onto M.

() lawful Kap-isomorphisms preserve Kap.

(h) [Density] For every in +, and M Kap, there is M Kap such

that M Kap M and |M|.

(i) [Amalgamation] Assume Ml Kap for l < 3 and M0 Kap Mlfor l = 1, 2. Then for some lawful function f and M Kap, we

have M1 Kap M, the domain of f is M2, the restriction f |M0|

is the identity, and f is a Kap-embedding of M2 into M, i.e.f(M2) Kap M. If M1 M2 = M0, we can assume that f = id.

Remark 1.5. (1) There is no contradiction concerning vocabularies in (g)()

of Definition 1.4(3): ifKap is a weak -approximation family, while M Kapand h is a lawful mapping from |M| onto some u, then (u) = (|M|) (so

saying that h gives rise to a Kap-isomorphism makes sense).

[Why? Letting def= sup(u), if < , we can find |M| such that

h() (, ). Hence

< + h() + = + < sup(|M|).

So, sup(|M|), and the other side of the inequality is shown similarly.]

(2) If M = Mi : i < i is a Kap-increasing sequence, and |, then

Mi : i < i is Kap-increasing, by Definition 1.4(1)(c), and if i

< ,i


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Notation 1.6.Suppose that Kap is a weak -approximation family and

is a sequence of vocabularies as in Definition 1.4(1)(a). We say that Kap is

written in .

Definition 1.7. [Sh 457]

(1) Let (Kap, Kap) be a weak -approximation family and Kap. We

say that is (< )-closediff for every Kap-increasing chain of size <

of elements of , the lub of the chain is in .

(2) Suppose that (Kap, Kap) is a weak -approximation family. We let

Kmd = Kmd[Kap]

def=

:

(i) is a (< )-closed subset of Kap,(ii) is Kap -directed,(iii) for cofinally many < + we have(M )( |M|)() = ()

(e.g. = )

.

We let Kmd = Kmd[Kap]def=

K

md :

(iv) (M & M Kap M1) =

(M2 )(h lawful)[h : M1 M2embedding over M]

(v) M & N M = N

.

(3) IfKap is as above and < +, we define Kmd[K

ap] as the set of Kap

such that

(a) M = |M| ,

(b) satisfies (i) (ii) from (2) above.

Similarly for Kmd[Kap].

Claim 1.8. Suppose that Kmd[Kap], while N and h : N M is a

lawful embedding. Then there is N and a lawful embedding g : M N

such that for x N we have g(h(x)) = x.

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Proof of the Claim.There is a lawful isomorphism f : M M

for someM N such that f(h(x)) = x for all x N. Then by (iv) in the definition

of Kmd, there is N and a lawful embedding g : M N such that

g N = idN.

Let g : M N be given by letting g(x) = g(f(x)), so g is a lawful

embedding and for x N we have g(h(x)) = g(f(h(x)) = g(x) = x. 1.8

Definition 1.9. (1) K = (K, K) is an abstract elementary class iff K is a

class of models of some fixed vocabulary = K and K=K is a two place

relation on K, satisfying the following axioms:

Ax 0: If M K, then all -models isomorphic to M are also in K. The

relation K is preserved under isomorphisms,

Ax I: IfM K N, then M is a submodel of N,

Ax II: K is a partial order on K,

Ax III, IV: The union of a K-increasing continuous chain M of elements

of K is an element of K, and the lub of M under K,

Ax V: If Ml K N for l {0, 1} and M0 is a submodel of M1, thenM0 K M1,

Ax VI: There is a cardinal such that for every M K and A |M|, there

is N K M such that A |N| and ||N|| (|A| + 1). The least such

is denoted by LS(K) and called the Lowenheim-Skolem number ofK.

(2) If is a cardinal and K an abstract elementary class, we denote by Kthe family of all elements of K whose cardinality is .

(3) For K an abstract elementary class, and a cardinal, we say that Khas a universal iff there is M K such that for all M K we have that

some M which is isomorphic to M satisfies M K M. Such M is called

universal for K.

(4) Suppose that K is an abstract elementary class. We shall say that a mem-

ber M ofK is K-embeddable in a member N ofK iff there is an isomorphism

between M and some M K satisfying M K N.

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(a) K is said to have the joint embedding property iff for any M1, M2 K,there is N K such that M1, M2 are K-embeddable into N.

(b) K is said to have amalgamation iff for all M0, M1, M2 K and K-

embeddings gl : M0 Ml for l {1, 2}, there is N K and K-

embeddings fl : Ml N such that f1 g1 = f2 g2.

Similar definitions are made to describe when K has the joint embedding

property or amalgamation.

Convention 1.10. We shall only work with abstract elementary classes

which have the joint embedding property and amalgamation.Note 1.11. The following notes are not hard and the proofs are to be found

in [Sh 88]. We include them here for the readers convenience.

(1) Suppose that K is an abstract elementary class. If M = Mi : i < is a

K-increasing chain (not necessarily continuous), then

i


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such that

(1) For all i, we have i,

(2) M Kap = M K,

(3) M Kap N = M K N and

(4) For every M K with ||M|| < there is N Kap such that

M is K -embeddable into N .

We say that Kap tends to [strongly] -approximate K.

We may just say Kap tends to approximate K if the rest is clear from

the context.

Observation 1.13. Suppose that Kap is a strong -approximation family

which tends to approximate K and Kmd. Then

(1) M defined by letting

Mdef=

MM

is an element ofK and for every M we have M K M, and in

fact M is the K-lub of {M : M }.

(2) For every , Kmd[Kap] such that , we have M K M .

[Why? (1) As {M : M } is K-directed.

(2) By (1) and Note 1.11(2).]

Notation 1.14. Suppose that an approximation family Kap tends to ap-

proximate K, while Kmd. If we write M, we always mean the model

obtained from as in Observation 1.13.

Definition 1.15. Let Kap be a strong -approximation family which tends

to -approximate K and let K+ be a subclass of K+ . Assume

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() For every M

K+

, there is K

md[Kap] with {|M| : M } a clubof [Ev]


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[ T i & i i

] = gi

= gi

.i = 0. Trivial.

i = i + 1. Let levi(T) = {j : j < j +}. For simplicity in notation

we assume that j is a limit 0, the other cases are similar. By induction

on j we build Mj : j < j so that M0 = M

[T i] and Mj , M

j K M

j+1,

while ||Mj || = , and M =

j


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(vi) Some h is a lawful Kap-isomorphism from N1 (1) onto N2 (1)mapping M1 onto M2,

(vii) h (N1 1) is the identity.

Then there are M and N Kap with M N, and gl for l {1, 2}

such that Ml M N and gl is a Kap-embedding of Nl into N, with

gl Ml = idMl. In addition, |N| {2 : < +} = |M| and gl (Nl l) is

fixed.

Note 1.19. For those familiar with definitions in [Sh 457], we emphasize

that smoothness was assumed throughout. That is, our definition of Kap isless general than the one in [Sh 457], and any strong -approximation family

in the sense of our Definition 1.7 automatically satisfies the condition which

in [Sh 457] was called smoothness.

2 Universals in +.

Definition 2.1. [Sh 546] Suppose that > 0 is a cardinal and < a limit

ordinal. A forcing notion Q satisfies iff player I has a winning strategy in

the following game [Q]:Moves: The play lasts moves. For < , the -th move is described by:

Player I: If = 0, I chooses qi : i < + such that qi Q and q

i p

i

for all < , as well as a function f : + + which is regressive on

C S+

for some club C of +. If = 0, we let qi

def= Q and f be

identically 0.

Player II: Chooses pi : i < + such that qi p

i Q for all i <

+.

The Outcome: Player I wins iff:

For some club E of +, for any i < j E S+

,


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(2) A winning strategy for I in

[Q] is a function St= (St, St

) such that inany play

qi : i < +, f, p

i : i <

+ : <

in which we have for all , i

qi = St(i,

pj : j < + : <

), f = St

(

pj : j < + : <

),

I wins.

i.e. a winning strategy for I depends only on the moves of II and f and C can be defined from

pj : j <

+ : < .

Fact 2.2. [Sh 546] Suppose that > 0 is a cardinal satisfying


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(D) If > 0, we are given a limit ordinal < .Then in V

def= VR

0 for some P we have

(a) P is a forcing notion of cardinality ,

(b) P is (< )-complete and +-cc (and if > 0, P satisfies ),

(c) In VP we have 0, then the following holds in VP

: if Q is a (< )-completeforcing notion of cardinality < and satisfies , and if we are given

a family {Ij : j < +} of dense subsets of Q, then for some directed

G Q we have that G Ij = for all j < +,

(e) In VP, if K = Kap is a workable strong -approximation family, then

we can find

= : T : <

++

such that

(i) For every < ++ and T we have Klg()ap is Kap-directed, and also for T, we have that

= {M ( lg()) : M },

(ii) For any +-branch of T and < ++, we have

{ : } K

md[Kap],

(iii) For any Kmd[Kap], for some < ++ we have that M is

isomorphically embeddable into M

i


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Once we prove the theorem, we shall be able to draw the following

Conclusion 2.4. Suppose that V satisfies

0 = 0, we are given a limit ordinal < . Then there is a cofinality

and cardinality preserving forcing extension V of V which satisfies

(1) For every abstract elementary class K for which there is a workable

-approximation family Kap which approximates K, and such that

LS(K) , there are ++ elements {M : < ++} of K+ which

are jointly universal for K+,

(2) 0 0: if Q i s a (< )-complete forcing notion of

cardinality < , satisfying , and we are given a family {Ij : j < +}

of dense subsets of Q, then for some directed G Q we have that

G Ij = for all j < +

,

(4) IfK is an abstract elementary class with LS(K) and K+ is a subclass

ofK+ for which there is a workable strong -approximation family Kapwhich approximates K+, and such that for every tree Tof the form from

Claim 1.16 in which every M is the union of elements of Kap we

have that M[T] K+, then there are ++ elements {M : < ++}

of K+ which are jointly universal for K+.

Remark 2.5. The informal plan of the proof of the theorem and the con-

clusion is as follows. The purpose of forcing with R is to make 2+

= andadd branches through T. Then P will be an iteration of ++ blocks of

steps each. Hence

P = P, Q

: ++, < ++

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and for each we have Q = Q

i , R

j : i , j < . Each R

j will be one offour possible kinds (three in case = 0). Let us first describe the situation

when > 0.

At kind 1 coordinates we shall be taking care of the form of Martins

Axiom given in (d) of the Theorem. Each kind 2 coordinate R

+1j takes a

workable strong -approximation family Kap from VP and a family of

elements ofKap[Kmd] and introduces a tree of elements of Kap indexed by T,

which gives as in (e)(i)-(ii) of the Theorem. This tree will also have the

property that for every Kap[Kmd] VP , there is a branch of T with

V and a tree T whose elements are pairs (N, h) with N and h an

embedding from N into Mi 0, are simplified

and guarantee (e)(i)-(iii) immediately. This eliminates the need for kind threecoordinates.

To get the conclusion for a given K as in (1), recall from 1 that for every

< ++ there is a model M in K+ such that for every branch of T,

the model read along the branch in the tree indexed by , embeds into

M. As Kap approximates K (see Definition 1.15), for every M K, there is

Kmd (and of the kind required by the Theorem) such that M embeds

into M . From the Theorem, there is Kmd such that M embeds into

M. This is in VP for some < ++, and hence some R

+1j will guarantee

that M embeds into M.

The proof of the Conclusionis given after the proof of the Theorem, close

to the end of the section.

Proof of the Theorem. Let R be as in the statement of the Theorem,

and let V = VR

. Then in V we clearly have 0 =


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2+

= and the cardinalities and cofinalities of V0 are preserved.Let f : < list the

+-branches of T in V.

We make some easy observations:

Note 2.6. (1) It suffices to prove the conclusion weakened by requiring each

Q being considered in (d)1, to have the set of elements some ordinal < .

(2) By renaming, each Kap considered in the theorem can be assumed to

have its vocabulary included in H(+).

Definition 2.7. We define P as P++ in the iteration

P = P, Q :

++

, <

++

,where

() P is a (< )-support iteration.

() For each < ++, in VP we have that Q is Q in the iteration

Q = Qi , R

j : i , j < ,

where:

(i) the iteration in Q is made with (< )-supports,

(ii) for each j < one of the following occurs:

Case 1. R

j is a Q

j -name of a (< )-complete forcing notion which

satisfies if > 0, and is ccc if = 0; and whose set of elements is

some ordinal < .

Case 2. For some P-name of a workable -approximation family K

ap,j,

abbreviated as K

, and elements { = ,j : < } ofK

VP

md , we have

that R

= R

j is defined as follows. We work in V

PQ

j . For M K we

let

w[M]def

= {, + 1 : M [,(+ 1)) = }.

Subcase 2A. = 0. The elements of R are conditions of the form

p = up, Mp : up, bp, cp : b

p, (Np,, hp,) : b

p, cp,

where

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(a)[closure under intersections] u = up

[T]


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(a)-(e) from Subcase 2A hold,(f)B for b

p, cp we have dp, []

0, then we are

given < and a P-name K

= K

ap,j of a workable -approximation

family such that for some j < j we have had K

ap,j = K

ap,j and the

forcing R

j was defined by Case 2. In V

PQ

jR

j , let G be the generic

of Rj over VPQ

j and let Rjdef=

{(N, h) : (p G)( cp)( dp,)[(N, h) = (N

p,,, h

p,,)]}

ordered by (N1, h1) (N2, h2) iff N1 N2 and h1 = h2 N1.Case 4. For some P Q

j -names of a workable -approximation family

K

= K

ap,j and a member

=

,j of K

md[K

ap] such that

PQ

j

{|M| : M } [Ev]


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we have (working in VP

Q

j ),

R =

M, N : M, N K & M = N Ev & M

ordered by

M1, N1 M2, N2 iff [M1 M2 and N1 N2].

Discussion 2.8. We now prove a series of Claims which taken together im-

ply the Theorem. These Claims are formulated for < ++, j < and are

proved by induction on and j. Let us fix < ++ and j < and assume

that we have arrived at the induction step for (, j). We work in VPQ

j

andlet R = Rj .

Claim 2.9. Suppose R is defined by Case 2 of Definition 2.7.

(1) If p R, then for any up and C up a chain with

C = , we have

Mp =

{Mp : C }.

(2) For every T and v [+] 0 and that p R, < and (N1, h1), (N2, h2) are

such that for some 1, 2 cp and l d

p,l

we have

(Nl, hl) = (Np,l,l

, hp,l,l) for l {1, 2},

while

h1 (N1 N2) = h2 (N1 N2).

Then

D = {q p : ( cq)( dq,)

l{1,2}

Nq,, Np,l,l, hq,, hp,l,l}

is dense above p.

(5) Suppose that > 0 and < . Then for every N , the set of all

p R such that for some , we have Np,, N is dense.

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Proof of the Claim.(1) Obvious.

(2) Clearly J,v is open, we shall show that it is dense. Given p R, we shall

define q J,v with q p. We do the definition in several steps.

Step I. Let udef= up {} { : up}.

For u we define M as follows. If for some up we have ,

then let Mdef= M ( lg()). Once this has been done, we have = / up

and we let

Mdef=

{M : u & }.

It has to be checked that this definition is valid, in particular that M is well

defined for for which there are 1 = 2 both in up

with 1 2. As up

is closed under intersections, we have in this case that def= 1 2 is in u

p

and

Mp1 ( lg()) = Mp = M

p2 ( lg()),

and hence M is well defined. Observe also that u is closed under intersections

and that

u = M = M ( lg()),

while clearly u. Also note that if up we have u = up and M = Mp

for u.

Step II. For u, we find M with M Kap, M and vlg() |M

|,

while |M| lg() and such that for u we have M = M

lg().

This is done by induction on lg(). Coming to , if =

{ u : },

let Mdef=

{M : u & }. Suppose otherwise and let

def=

{ lg() : u & } and Mdef=

{M : u & }.

Since M M by the inductive assumptions, by the axiom of end

amalgamation in Kap we have that M and M are compatible and have a

common upper bound M

with the property lg() |M

| and such that

for u with , we have that M

lg() = M. Ifv lg() |M

|

let M

= M

. Otherwise, let f be a lawful embedding of M

into M

with

f = id and (M

\ ) (M

\ ) = , while |M

| (v lg()) \ M

.

Applying amalgamation to M

, M

we can find M

as required.

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Step III. Now for u let Mq

def

= M

. Let

uqdef= u { : u & w[Mq]}.

For uq, let Mqdef= M if u, and otherwise let M

q = M

q ( lg())

for any u with .

Subcase A. = 0. Let

qdef= uq, Mq : u

q, bp, cp : bp, (Np,, h

p,) : b

p, cp.

Subcase B. > 0 Let

qdef= uq, Mq : u

q, bp, cp : bp, dp, : b

p, cp,(N, h)

p.

It is easily seen that q is as required.

(3) All coordinates ofq will be the same as the corresponding coordinates of

p, with the possible exception of up and Mp : up. Let u = up. For

u we define Mq Mp by induction on lg(). The inductive hypothesis

is that if , then Mq = M lg(). The proof is similar to that of of

Step II of (2). Coming to , let and M be defined as there. If , then

we let Mq = M lg(), and we have Mq M

p by the choice of M.

Otherwise, let = , and hence u and lg() < lg(). We have

that

Mq = M lg() Mp = M

p lg().

Hence we can find Mq Mp with M

q lg() = M

q by end amalgamation.

Once the induction is done, we define uq exactly as in the Step III of (2),

and define Mq for uq accordingly.

(4) Let r p, we shall find q D with q r. For some 1, 2,

1,

2 we have

N1 Nr,1

,1

and N2 Nr,2

,2, and similarly for h1, h2. For simplicity of

notation, we assume l = l and l = l for l {1, 2}. Let = max{1, 2}

and let = f . Let M def= Mr , which is well defined. Hence h1 and h2

are lawful embeddings of N1 and N2 into M respectively.

First define a lawful isomorphism g0 from M onto some M1 such that

for x Nl we have g(hl(x)) = x for l {1, 2}. This is possible because h1and h2 agree on N1 N2. Hence we have that N1 and N1 M1. As

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Kmd, there is a lawful embedding g1 : M1 M2 for some M2 suchthat g1 is the identity on N1. Without loss of generality, again as Kmd,

we can assume that |M2| .

Now let g2 be a lawful isomorphism between M2 and M3 such that

g2 N1 = id and g2(g1(x)) = x for x N2. Then N2 M3, so we can find

M4 and a lawful embedding g3 : M3 M4 over M3. Then N1, N2 M4.

Without loss of generality we have |M4| . Finally, there is a lawful

isomorphism g between M4 and some M such that g Nl = hl for l {1, 2}

and M M. By (3) we can find q r such that Mq

= M. We shall define

q q so that all the coordinates of q are the same as the corresponding

coordinates of q, except that we in addition choose some

\ (dq

,

{dq,j : j w[M4]}

and let Nq,, = M4, while hq,, = g. We let N

q,j, = M4 j for

j w[M4], and similarly for hq,j,. Then q is as required.

(5) Similar to (4), using (3) and (4). 2.9

Claim 2.10. (1) If > 0, then R is a < -complete forcing.

(2) If R was defined by one of the Cases 3-4 or by Case 2 and = 0, thenevery increasing sequence in R of length < has a least upper bound.

(3) Suppose R was defined by Case 2. Then, for every <


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Suppose that q = qi : i < i

< is an increasing sequence in R.Without loss of generality, i is a limit ordinal. Let bdef=

i


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Note that D VPQ

j

as all forcings involved are (< )-closed by the induc-tion hypothesis.

Subclaim 2.11. D is dense above p in the forcing Rj.

Proof of the Subclaim. Let r p be given. For every up with fwe have Mr M

p. Hence if u

r we have Mr =

{Mr : } M. By

Claim 2.9(2) we can without loss of generality assume that this is the case.

Let uq = u and for uq let Mq = Mr.

Let = lg(), and hence |M| . Further let cq = cr {}. Let be

such that /

dr, and for

with

cq let

(Nq,,, hq,,) = (N

, h ).

We complete the definition of q in the obvious fashion. Hence q r and

q D. 2.11

By the Subclaim it follows that there is q G D, and this q witnesses

that (N, h) R. Obviously, (N, h) is the lub of (Ni, hi) : i < i.

Case 4. If Mi, Ni : i < i < is increasing in R then clearly

i


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(< )-closed subset of Kap and it is equally easy to see that it is directed.To see that

{|M| : M

i 0, then R satisfies .

If = 0, then R satisfies ccc.

Proof of the Claim. We distinguish various cases of Definition 2.7.

Case 1. R is defined by Case 1 of Definition 2.7. The conclusion follows

by the assumptions.

Case 2. (main case) R = Rj is defined by Case 2 of Definition 2.7. As

Subcase B is more difficult, we start by it.Subcase B. > 0. Let K = K,jap , and let us follow the rest of the

notation of Definition 2.7 as well. By our assumptions we have |T | = + and

by Claim 2.10, the equality (+)


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Proof of the Subclaim.For N K define

o[N]def= { : w[N]} |N|,

[N]def= min{ < : = o[N] = g1() = g

1() },

Ndef= {g1 () [N] : o[N]}.

Note that [N] is well defined because g1 is 1-1 and |o[N]| < .

For < +, let g : be one to one. For N K let AN be a model

with universe included in N such that the function

g1() [N]

is an isomorphism from N onto AN. Let


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and hence f1() > f2(), which means that g

1() >N1 g

1() ,a contradiction. A similar contradiction can be obtained by assuming that

g() < .

If N1 then g() < g() = . By the definition of g we

have g1() = g1(g()) . Hence,

, g() o[N2] and g() < . So

(g1(g()) , g1(

) , g(g())) RN2 = RN1. As also

(g1() , g1(

) , g()) RN1,

we have that g(g()) = g() and hence g() = . In particular

o[N2]. As we have N1, we have g1() AN1, and hence

g1() = g1() = g

1(g()) for some N2. As = [N2], we

have that = g() = , so N2 .

This argument shows that N1 N2

, and it can be shown

similarly that N1 = N2

as sets and as models. 2.15

For p R, let ((p,i) : i < i(p) list up with no repetitions, and let (p)

be the minimal < such that

g1(g0((p,i))) : i < i(p)

is without repetitions (which exists as g0 and g1 are 1-1 and ((p,i) : i < i(p)

is without repetitions). Let

g3 : R

be such that for p, q R with g3 (p) = g3(q) we have

(a) i(p) = i(q),

(b) the mapping defined by sending (p,i) (q, i) preserves

, ( ), 1

2

= , (1

2

= ),

(c) (p) = (q),

(d) for i < i(p) we have g1(g0((p,i))) (p) = g

1(g

0((q, i))) (p) (recall

that |>2| = ),

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(e) for i < i(p) we have g

2(M

p

(p,i)) = g

2(M

q

(q,i)).

The existence of such a function can be shown by counting.

Subclaim 2.16. Ifg3(p) = g3 (q), then the mapping sending (p,i) to (q, i)

for i < i(p) = i(q), is the identity on up uq.

Proof of the Subclaim. Suppose that up uq. Let i be such that

= (p,i). Letting def= (p) = (q),we have

g1(g0()) = g

1(g

0((q, i))) .

By the definition of and the fact that uq, we must have (q, i) = .

2.16

Let us also fix a bijection

F : >([+]([j]


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Let us check that this definition is as required. It follows from the choice ofC that each f is regressive on C S

+

. Let E C be a club of + such

that

[j E S+

& j < j] = ( < )(i < i(qj)) [w[M

q

j

(qj

,i)] j].

Suppose that j < j E S+

are such that


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such that k uqk

lk for k {1, 2}, and Mq1

l11 lg() = Mq2

l22 lg().By taking the larger of 1, 2, we may assume that 1 = 2 = . By the

closure under intersections, we can also assume that l1 = l2, so without loss

of generality we have l1 = j and l2 = j. Let lg() be minimal such that

Mqj

1 = Mqj

2 .

By the minimality of , we have = + 1 for some w[Mq

j

1 ] w[Mqj

2 ].

As j E we have that w[Mq

j

1 ] j, so w[Mqj

2 ] j. As f(j) = f(j),

there is uq

j

such that for some i < i(q

j ) = i(q

j) we have 2 = (q

j , i),

= (qj, i) and w[Mqj

2 ]j = w[Mqj

]j. Hence we have g2(Mqj

) = g2(Mqj

2 ),

and as , w[Mqj

] w[Mqj

2 ], we have Mqj

= Mqj

2 . We have not

arrived at a contradiction yet, as we do not know the relationship between

and 1.

As lg(), we have def= 1 = 2 . Since w[M

qj

1 ], we have

= 1 uqj and similarly uq

j . Let o be such that = (qj, o). By

Subclaim 2.15 we have = (qj , o). Since we have 2 by the choice of

g3, we have . So 1 , and as we have 1 uq

j , we obtain

Mqj

1 = Mqj

1 lg() = Mqj

lg() = Mqj

2 lg() = Mqj

2 ,

a contradiction. This proves the first part of the statement. If u and

l {j,j} is such that uql, then we have M = Mql , as is clear from the

definition. 2.17

Now we let

urdef= u {w[M] : u},

and define Mr for u

r

accordingly, which is done as in Step III of theProof of Claim 2.9(2).

We let br = bqj bqj and for b we let cr = cqj c

qj . For b, cr

we let dr, = {2 : dqj,} {2 + 1 : d

qj,} and

(Nr,,2, hr,,2) = (N

qj,,, h

qj,,) while (N

r,,2+1, h

r,,2+1) = (N

qj,,, h

qj,,).

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We have now completed the proof of Subcase B of Case 2 of the Claim.Subcase A. = 0. We have to prove that R satisfies ccc. Let functions

g0, g1, g

2 and g

3 be as in the proof of Subcase B, and let the function F and

the club C be given as in that proof.

Suppose that we are given a sequence qs : s < 1 of conditions in R.

Let E C be a club of1 such that

j S10 & j < j = (i < i(q, j))(w[M

qj

(qj ,i)] j).

Let g be a function that to an ordinal j S10 assigns

(g3(qj ), w[M

qj(qj ,i)] j : i < i(qj ))

and let

f = (F g) (E S10 ) 01\(ES10 ).

Exactly as in Subcase B, it follows that whenever

j < j S10 E & f(j) = f(j)

then letting

u = uqj uqj { : uqj , uqj}

and for u

M = Mql lg()

for any l {j,j} for which uql , we obtain a well defined sequence

M : u of elements of K with the property that for any l {j,j},

uql we have Mql M. Let S S10 E be stationary such that f(j)

is fixed for j S. We apply the -system lemma to {bqj : j S} and

obtain A [S]1 and b [] 0. Using the -system lemma n times if

necessary, we can find B [A]1 and for b a set c [1]


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(i) c

qj

c

qj

= c

,

(ii) min(cqj \ c) > max{ : c

},

(iii) min(cqj \ c) > max{ : c

qj },

(iv) c = (Nqj, , h

qj,) = (N

qj,, h

qj,)

and for k < ndef= |c| letting

k, k be the k-th element of c

qj , c

qj respec-

tively, we have that Nqj

,k

and Nqj,k are isomorphic. Let j

< j B and let

b. Let Nj = cqj

Nqj, and Nj = c

qj

Nqj,, while hj = c

qj

hqj,

and hj =

cqj hqj,. Then Nj and N

j are isomorphic and hj and hj agree on

their intersection, while there are 0 < 1 < 2 divisible by such that

Nj 0 = Nj 1 = Nj N

j

with |Nj| 1 and |Nj| 2 and = f 2 u

qj . We also have that

Nj, Nj . Then hj is a lawful embedding of Nj into M and hj is a lawful

embedding ofNj into M, by the choice ofS. Similarly to the proof of Claim

2.9(4), we can see that qj and qj are compatible, by finding N with

N N1, N2, extending qj, qj to enlarge M and then taking an upper bound

of the extensions.Case 3. Suppose that

(qi : i < +, f), p

i : i <

+ : <

have been played so far. By Claim 2.10 we can have qi be the lub of

qi : < . Let I : < list the isomorphism types of elements of

K. Let F be a bijection

F : K {h : h a lawful function with Dom(h) [+]


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Then let f = (F g) (C S+

) 0+\(CS+ ). Let E C be a club of+ such that

j E S+

& j < j = ( < j )(|N

qj | j).

Let (N, h) be the lub of {(Nqj , h

qj ) : < } and (N, h) the lub of

{(Nqj , hq

j ) : < }. We shall show that (N, h) and (N, h) are compati-

ble. As N, N , clearly they are compatible as elements of K. We need

to show that h and h agree on N N.

Suppose not and let < be the least such that h and h disagree

on Nq

j Nq

j -such a exists by the definition of the lub in the forcing.By the choice of E we have |N

q

j | j and by the choice of f we have

(Nq

j j, hq

j (Nq

j j)) = (Nqj j,hq

j (Nq

j j)). Hence N

q

j Nqj j

and hq

j and hqj agree on this intersection, a contradiction. We can also see

that N and N are isomorphic.

Now let G be as in the Definition 2.7 Case 3 and let p G witness that

(N, h) R, while p G witnesses that (N, h) R. Let p+ G be a

common upper bound of p and p. By Claim 2.9(4) it follows that

D = {p++

p+

: ( cp++

)( dp++

, ) (N, N

Np++

,, & hh

hp++

,,)}

is dense in the forcing R giving rise to G, which suffices.

Now suppose that we are in Case 4 and that(qi : i <

+, f), pi : i <

+ : <

have been played so far in the game [R]. This is where we get to use the

workability of K. As before, we let player I choose qi as the unique least

upper bound ofpi : < . Let qi = M

i , N

i . Using =


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At the end, let C +

be a club such that for every <

i < j & j C = |Ni | j.

Suppose now that i < j C S+

are such that f(i) = f(j) for all < .

For l {i, j} let Mldef=


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can apply amalgamation to N , M , M

to find N

Kap N with M

N

. ByRemark 1.5 (3), we can assume that N Ev = M. Hence M, N R I

is as required, showing (iii).

To show (iv), suppose N and N Kap N after we have forced by R.

As the forcing with R is (< )-closed, we have N VPQ

j and N Kap N

holds. Let M be such that M, N GR. Now observe that by amalgamation

and Remark 1.5(3), the set

{M, N : ( lawful h)[h : N N embedding over M]}

is dense in R above M, N. 2.18

This finishes the inductive proof.

Claim 2.19. It is possible to define the iteration P so that in VP we have

(1) If > 0 then for every (< )-complete forcing notion Q which satisfies

and has the set of elements some ordinal < and for every < ++

large enough we have R

j = Q for some j < . If = 0, the analogous

statement holds with ccc forcing in place of (< )-complete forcing,

(2) For every workable strong -approximation family K and a family

= { : < } of elements of Kmd, and for every < ++ large

enough, there is j < such that Rj is given by Case 2 of Definition

2.7 using K, as parameters.

(3) If > 0, then for every K, , , j as in (2), for every < , there is

> such that R

j is defined by Case 3 of Definition 2.7 using and as parameters.

(4) For every workable strong -approximation family K and K

md

[K]

such that

{|M| : M } Ev, for every j large enough there is

< ++ such that Rj is defined by Case 4 of Definition 2.7 using

as a parameter.

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Proof of the Claim.We use the standard bookkeeping. As the forcing is

(< )-closed, any workable strong -approximation family K VP appears

at some stage and does not gain any new members later. Also notice that

being in Kmd and Kmd is absolute between VP and VP

jQ

j containing for

K. 2.19

2.3

This finishes the proof of the Theorem.

Remark 2.20. Applying the usual proof of the consistency of MA + CH

if we assume in Theorem 2.3 that V satisfies

< =


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(2) In addition to what we have already observed, we need to observe that2 = , and this is the case because P adds a Cohen subset to many

times.

(3) Follows from (1) of the Theorem.

(4) This part follows similarly to (1), using the assumptions on K+. 2.4

Fact 2.21. Suppose = I, we define a functional ft : BI R

by letting

ft(iI

aixi)def=

l


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LetF

def= {ft : t

n


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Definition 3.5.(1) We define an iteration

P, Q

: , <

with (< )-supports such that for all < we have that Q

is a P-name

defined by

Q

def= {(w

,

w) : w

[+]


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For j < +

, let (j,n)

def

= sup(wn

j j). Note that for j S+

we have(j,n) < j. Let Cn+1

def= E. Define gn+1 which to an ordinal j S

+

assigns

(wnj j, nj (w

nj j), (j,n)).

Then let

fn+1def= (F gn+1) (Cn+1 S

+

) 0+\(Cn+1S+ ).

Hence fn+1 is regressive on Cn+1 \ {0}.

At the end of the game, for i < + let widef= n


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Without loss of generality, p

decides the value c of c. Let 0 < n

< besuch that c < n. Let x

i for i <

+ be the generators of BI

. For i < + we

find pi P/P such that p pi and

pi h

(x

i) = yi for some yi.

Let us now work in VP. Let pi() = (wi,


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Proof of the Observation.The first statement follows by comparison with

11

(x+1)ln(x+2)dx. The second statement follows from the following estimate:

nl=1

1(nl+1)ln(l+2)

[n/2]

l=11

(nl+1)ln(l+2)+n

l=[n/2]+11

(nl+1)ln(l+2)

1[n/2]+1

[n/2]l=1

1ln(l+2)

+n[n/2]

l=11

l ln(nl+3)

1[n/2]+1

[n/2] 1ln 3

+ 1ln([n/2]+2)

n[n/2]l=1

1l

1

ln 3+ 1

ln([n/2]+2)(1 +

n[n/2]1

1x

dx) 1ln 3 +1

ln([n/2]+2) (1 + ln(n [n/2]))

1ln 3 +1

ln 2+ 1 < 4.

3.7

By Observation 3.7, we can choose m large enough such that

ml=1

1

(l + 1) ln(l + 2) 4(n)2.

Let us choose i1 < < im Y.

Claim 3.8. We can find q and q in Q, both extending all pil() for

1 l m, and such that

q Q i1, . . . , im is increasing in I

and

q Q i1, . . . , im is decreasing in I

.

Proof of the Claim. Notice that for no 1 l1 < l2 m and i Y do

we have that pi decides the order between il1 and il2 in I, by the choice

of Y (this is elaborated below). The proof can proceed by induction on m.

The inductive step is as in the proof of . The only constraint we could

have to letting il1 il2 (for q) or il2 il1 (for q

) would be if some z w

would prevent this, but this does not happen. For example, if we could not

let il1 il2 in q then this would mean that il1 il2 would have to hold. By

the choice of Y and since il1 wil1 \ w and similarly for il2, this could only

be the case if for some z w it would hold that il1 wil1z while il2 wil2

z.

However, this would contradict item (d) in the choice of Y. 3.8

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Back in VP

, let z

def

=m

l=1

1

l+1 xil . Let a

def

= ||m

l=1

1

l+1 yil ||A. Hence

q ||z

||BI

4(n)2,

and so a 4(n)2/n = 4n. On the other hand,

q ||z

||BI

< 4,

and hence a < 4n, a contradiction. 3.4

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