On potential isomorphism and non-structure

Digital Object Identifier (DOI):10.1007/s00153-003-0185-z

Arch. Math. Logic 43, 85–120 (2004) Mathematical Logic

Taneli Huuskonen · Tapani Hyttinen ·Mika Rautila

On potential isomorphism and non-structure

Received: 15 March 2001 / Revised Version: 6 September 2002Published online: 9 September 2003 – © Springer-Verlag 2003

Abstract. We show that for any non-classifiable countable theory T there are non-iso-morphic models A and B that can be forced to be isomorphic without adding subsets ofsmall cardinality. By making suitable cardinal arithmetic assumptions we can often preservestationary sets as well. We also study non-structure theorems relative to the Ehrenfeucht-Fraısse game.

1. Introduction

J.T. Baldwin, M.C. Laskowski and S. Shelah point out in [2, p. 1291–1292] that itis not possible to force two non-isomorphic models of any classifiable theory to beisomorphic without collapsing cardinals or adding new countable sets. They alsopoint out that for any non-classifiable theory T there are non-isomorphic modelsthat can be forced to be isomorphic by a c.c.c. partial order. This can be interpretedas a non-structure theorem, since the result shows that any invariant that is pre-served by c.c.c. partial orders cannot be sufficient to characterize the models ofT .

In this paper we take a closer look at what happens if T is not classifiable.We aim, of course, at much stronger results than the one mentioned above. Wemeasure the strength of the result by how little the forcing notion changes theuniverse.

The problem of forcing non-isomorphic models to be isomorphic without add-ing subsets of small cardinality is also studied by M. Nadel and J. Stavi in [12]. Theycall non-isomorphic models λ potentially isomorphic if there is a partial order thatforces the models isomorphic without adding subsets of cardinality < λ. The mainresult in [12] is that for uncountable λ the relation≡∞λ cannot be characterized interms of potential isomorphism.

We also study more classical non-structure theorems in which the strength ismeasured by the length of Ehrenfeucht-Fraısse game in which player ∃ still has a

T. Huuskonen, T. Hyttinen, M. Rautila: Dept. of Mathematics, P.O.Box 4, 00014 Universityof Helsinki, Finland. e-mail: [email protected]

The research of the first and second author was partially supported by Academy of Finlandgrant 40734

Mathematics Subject Classification (2000): 03C55, 03C45

Key words or phrases: Potential isomorphism – Ehrenfeucht-Fraısse game

86 T. Huuskonen et al.

winning strategy. The non-structure theorems of this kind are closely related to theones mentioned above, as we shall see in this paper.

In [5, Theorem 4.4] it is shown that if λ is a successor of a regular cardinal µ

and the ideal I [λ] is improper then for any (λ+, ω+ 2)-trees I0 and I1 it holds that

I0 ≡λµ·ω+1 I1 iff I0 ∼= I1.

Here we show that the assumption “I [λ] is improper” cannot be removed from thetheorem. This is related to another problem, which had its 10th anniversary sometime ago: Assume λ > ω is a regular cardinal. Can we find non-isomorphic modelsA and B of cardinality λ such that they are rather strongly equivalent, e.g., player ∃has a winning strategy in the Ehrenfeucht-Fraısse game of length α for all α < λ?As far as we know, without the assumption that λ<λ = λ, no one has succeededin showing the existence of such models. In Section 6 we show that in the class of(λ+, ω+2)-trees it is equiconsistent with the existence of a Mahlo cardinal to havesuch equivalent trees I0 and I1 and λ<λ > λ.

2. Basic definitions

Our notation is fairly standard and can be found in almost any text book in settheory. Below we try to describe what we felt might not be standard enough.

Let p be a pair. The first component of p is denoted by 1st(p) and the secondcomponent by 2nd(p). The length of a sequence s is denoted by len(s).

Let λ be an ordinal and κ < λ a regular cardinal. We denote the set of ordinalsless than λ of cofinality κ by Sλ

κ , and the set of ordinals of cofinality < κ by Sλ<κ .

By Sλreg and Sλ

in we denote the set of regular cardinals and the set of inaccessible

cardinals less than λ, respectively. Similarly, Sλlim denotes the set of limit ordinals

less than λ. By LIMITS we denote the class of all limit ordinals. For a set X thecollection of subsets of X of cardinality strictly less than κ is denoted by Pκ(X).The club filter on κ is denoted by Dκ .

A set S ⊆ λ is a κ-club in λ if and only if it is unbounded in λ and κ-closed,i.e., every κ-cofinal ordinal δ < λ in which S is unbounded is in S. The set S isκ-stationary in λ if and only if it intersects every κ-club subset of λ. The followingfact is worth noting: A set S ⊆ λ is κ-club (κ-stationary) in λ if and only if there isa club (stationary) subset S′ of λ such that S ∩ Sλ

κ = S′ ∩ Sλκ .

Let A be a model. The vocabulary of A is denoted by τ(A). Suppose a ∈ Aand X ⊆ A. The submodel of A generated by X is denoted by hull(A, X). Bytpqf(a, X, A) we denote the quantifier free type of a over X in A. The set of partialisomorphisms from A to B is denoted by Part(A, B).

In connection to forcing we say that a condition p is stronger than a conditionq if p ≤ q. Some authors define this the other way around. When we use a tree U

as forcing notion, we mean that the ordering of the tree is reversed.Let I0 and I1 be linear orders. The sum of the linear orders is denoted by I0+I1

and the product is denoted by I0 · I1. We assume that the universe of the productI0 · I1 is I0 × I1.

On potential isomorphism and non-structure 87

2.1. Trees

Definition 2.1. (i) A tree 〈U,≤U 〉 is a partial order such that there is a leastelement, which is denoted by root(U), and for every x ∈ U , the set pred(x) ={y ∈ U | y <U x} is well-ordered by ≤U .

(ii) A branch of a tree U is a maximal chain of U . The set of branches of U isdenoted by br(U). Suppose x, y ∈ U . If pred(x) = pred(y), then we writex ∼ y. The equivalence class of x for ∼ is denoted by [x]. If pred(x) ∪ {x} =pred(y), then y is an immediate successor of x. By succ(x) we denote the setof immediate successors of x in U . The height (or level) of x, ht(x), is theorder-type of pred(x). The set of elements at level α is denoted by LevU(α).

(iii) Let λ be a cardinal and α an ordinal. By a (λ, α)-tree U we mean a tree thatsatisfies:

For all x ∈ U , |[x]| < λ.There are no branches of length ≥ α in U .If x, y ∈ U , x and y have no immediate predecessors and x ∼ y, thenx = y.

A tree U that satisfies only the last two conditions is called an α-tree.

The sum and product operations on trees as well as the supremum and infimumof a set of trees are fairly well standardized, but in order to avoid unnecessaryconfusion, we define the operations below.

Definition 2.2. Let U1 and U2 be trees, and {Ui | i ∈ I } a set of trees.

(i) The tree-sum of U1 and U2, denoted by U1 + U2, is the tree 〈U,≤〉 such thatU = U1× {0} ∪ (br(U1)×U2)× {1}, and for t1, t2 ∈ U , define t1 ≤ t2 if andonly if any of the following conditions holds:

t1 = 〈t ′1, 0〉 and t2 = 〈t ′2, 0〉 for some t ′1, t′2 ∈ U1 with t ′1 ≤ t ′2.

t1 = 〈t ′1, 0〉 and t2 = 〈〈S, t ′2〉, 1〉 for some t ′1 ∈ U1, t ′2 ∈ U2 and S ∈ br(U1)

with t ′1 ∈ S.t1 = 〈〈S, t ′1〉, 1〉 and t2 = 〈〈S, t ′2〉, 1〉 for some t ′1, t

′2 ∈ U2 and S ∈ br(U1)

with t ′1 ≤ t ′2.(ii) The tree-product of U1 and U2, denoted by U1 · U2, is the tree 〈U,≤〉 such

that U = {〈g, t1, t2〉 | t1 ∈ U1 ∧ t2 ∈ U2 ∧ g : pred(t2) → br(U1)} orderedby 〈g, t1, t2〉 ≤ 〈g′, t ′1, t ′2〉 if and only if any of the following holds:

t2 = t ′2, g = g′ and t1 ≤ t ′1.t2 ≤ t ′2, g = g′�pred(t2) and t1 ∈ g′(t2).

(iii) The supremum of Ui with respect to order-preserving embeddability, denotedby

⊕i∈I Ui , is the tree

U =⋃

i∈I

((Ui\{root(Ui)})× {i}

) ∪ {〈I, I 〉}

ordered by 〈t1, i1〉 ≤ 〈t2, i2〉 if and only if i1 = i2 and t1 ≤ t2, or i1 = I .


(iv) The infimum of the trees Ui with respect to order-preserving embeddability,denoted by

⊗i∈I Ui , is the tree

U = {f : I →

⋃

i∈I

Ui | ∃α∀i ∈ I(f (i) ∈ LevUi

(α))}

ordered coordinate-wise.

2.2. Games

All the games we consider are played by two players, whom we denote by ∀ and∃. It is convenient to assume that player ∀ is male and player ∃ is female.

There are two games of special importance for us, the club game for S and theEhrenfeucht-Fraısse game between models A and B. Below we describe the rulesof the games.

2.2.1. The definition of the club game Suppose that λ is an ordinal, A ⊆ Scf(λ)reg

and P a partial order. For a subset S of λ we define the A-club game for S of lengthP as follows. The game lasts at most cf(λ) rounds. On each round ξ < cf(λ) of aplay, first player ∀ chooses an element pξ ∈ P which is greater than all previouslychosen elements from P and an ordinal xξ < λ greater than all previously chosenordinals. Then player ∃ chooses an ordinal yξ < λ greater than xξ . The play endswhen cf(λ) rounds have been played or when player ∀ cannot follow the rules ofthe game anymore. At the end of the play, sequences x = 〈〈pξ , xξ 〉 | ξ < γ 〉 andy = 〈yξ | ξ < γ 〉 have been chosen where γ ≤ cf(λ) is the length of the play. Thepair 〈x, y〉 is known as the play. Player ∃ wins the play if sup{xξ | ξ < ζ } ∈ S forevery ζ ≤ γ with cf(ζ ) ∈ A. The game is denoted by GCA

P (S, λ). We denote the

game GC{µ}P (S, λ) by GCµP (S, λ), and GCµ

µ(S, λ) by GCµ(S, λ).

2.2.2. The definition of the Ehrenfeucht-Fraısse game Suppose that models Aand B are of the same similarity type. Let P still be a partial order and λ a cardinal.The Ehrenfeucht-Fraısse game of length P between models A and B with movesof cardinality strictly less than λ, denoted by EFλ

P (A, B), is played as follows. Onround ξ player ∀ first chooses an element pξ ∈ P greater than all previously chosen

elements from P . Then he chooses a cardinal κξ < λ and elements {xξα | α < κξ }

either from the model A or B. Then player ∃ chooses elements {yξα | α < κξ }

from the other model. Let {aξα | α < κξ } and {bξ

α | α < κξ } denote the elementschosen from the models A and B, respectively. Player ∃must choose the elements{yξ

α | α < κξ } so that the mapping

aζα �→ bζ

α, ζ ≤ ξ, α < κζ

is a partial isomorphism from the model A to B. The player who first must breakthe rules loses the play.

We write A ≡λP B if and only if player ∃ has a winning strategy in EFλ

P (A, B).


2.3. Good subsets of a regular cardinal

Everything in this section is due to Shelah.

Definition 2.3. Let λ be a regular cardinal. A subset S of λ is good if and only ifthere are a club subset C of λ and a sequence a = 〈aξ | ξ < λ〉 with aξ ∈ Pλ(λ)

such that for each α ∈ C ∩ S,

(i) α is a singular limit ordinal, and(ii) for some unbounded subset e of α of order type cf(α) it holds that

∀β < α∃ξ < α(e ∩ β = aξ ).

The collection of the good subsets of λ is denoted by I [λ].

The following lemma due to Shelah [16, Lemma 2] gives a few equivalentdefinitions of the ideal of good subsets of λ.

Lemma 2.4. Suppose λ is a regular cardinal and S ⊆ λ. Then the following areequivalent:

(i) S ∈ I [λ].(ii) There are a sequence 〈aξ | ξ < λ〉 with aξ ∈ Pλ(λ), and a set C ⊆ Sλ

lim clubin λ such that

∀γ ∈ S ∩ C ∃e ⊆ γ(sup(e) = γ ∧ otp(e) < γ ∧ cf(γ ) < γ ∧

∀α < γ ∃ξ < γ (e ∩ α = aξ )).

(iii) There are a sequence 〈Pξ | ξ < λ〉 with Pξ ⊆ Pλ(λ) of cardinality less thanλ, and a set C ⊆ Sλ

lim club in λ such that

∀γ ∈ S ∩ C ∃e ⊆ γ(sup(e) = γ ∧ otp(e) = cf(γ ) < γ ∧

∀α < γ ∃ξ < γ (e ∩ α ∈ Pξ )).

(iv) There are a sequence 〈Pξ | ξ < λ〉 with Pξ ⊆ Pλ(λ) of cardinality less thanλ, and a set C ⊆ Sλ

lim club in λ such that

∀γ ∈ S ∩ C ∃e ⊆ γ(sup(e) = γ ∧ otp(e) < γ ∧ cf(γ ) < γ ∧

∀α < γ ∃ξ < γ (e ∩ α ∈ Pξ )).

Proof. We prove only (ii) ⇒ (i). As to other non-trivial implications it suffices tonote that for any sequence a = 〈aξ | ξ < λ〉 of sets of cardinality strictly less thanλ there are a sequence b = 〈bξ | ξ < λ〉 enumerating the elements of the elementsof a and a set C club in λ such that for all α ∈ C and x, x ∈ b�α if and only if thereis ξ < α with x ∈ aξ .

Suppose S ⊆ λ, and let 〈aξ | ξ < λ〉 and C be as in (ii). For every limit ordinalδ < λ, choose club subset Cδ of δ of order type cf(δ), and for every ξ < λ, let

aξ,δ = {α ∈ aξ | otp(aξ ∩ α) ∈ Cδ}.


Let b = 〈bζ | ζ < λ〉 enumerate the set {aξ,δ | ξ < λ ∧ δ ∈ Sλlim}. Let

C0 = {α ∈ lim(C) | ∀ξ < α ∀δ ∈ Sαlim ∃ζ < α(bζ = aξ,δ)}.

Clearly C0 is a club subset of λ.Suppose that γ ∈ C0 ∩ S. Since C0 ⊆ C, there is e unbounded in γ of order

type less than γ . Let

e∗ = {α ∈ e | otp(e ∩ α) ∈ Cotp(e)}.Clearly otp(e∗) = cf(γ ). Suppose that α < γ . So there is ξ < γ with e ∩ α = aξ .By the definition of e∗ and aξ,otp(e), we have e∗ ∩ α = aξ,otp(e). Since γ ∈ C0,ξ < γ and otp(e) < γ , there is ζ < γ with bζ = aξ,otp(e). Thus, b and C0 witnessthat S ∈ I [λ]. ��

The following is well known.

Lemma 2.5. Assume ��κ . Then I [κ+] is improper.

Proof. Let λ = κ+, and let 〈Cξ | ξ ∈ Sλlim〉 be a box sequence. It suffices to show

that S = {α < λ | cf(α) > ω ∧ α > κ} ∈ I [λ]. Let

Pα ={{Cα ∩ β | β ≤ α} if α ∈ Sλ

lim

∅ otherwise.

Clearly |Pα| < λ for every α < λ. Let δ ∈ S be a limit ordinal. Suppose cf(δ) =µ > ω. Let 〈δξ | ξ < µ〉 be an increasing sequence cofinal in δ such that δξ ∈lim Cδ . Then for every ξ < µ, Cδξ = Cδ ∩ δξ . Suppose γ < δ. Let ζ = min({ξ <

µ | γ < δξ }). Since Cδ ∩ δζ = Cδζ , Cδ ∩ γ = Cδζ ∩ γ . Since Cδζ ∩ γ ∈ Pδζ ,δζ < δ and otp(Cδ) < δ, the sequence 〈Pα | α < λ〉 and the sets Cδ , δ ∈ S, witnessthat S ∈ I [λ]. ��

3. General non-structure theory

The general non-structure theory presented in this section is due to Shelah.

Definition 3.1. Let I and M be models, and � a collection of formulas in thevocabulary of M.

(i) The �-type of a in M is {ϕ(x) ∈ � | M |= ϕ(a)}.(ii) A collection 〈as | s ∈ I 〉 is �-indiscernible in M if for every n < ω, the

�-type of as1 � . . . � asn in M depends only on the quantifier free type of〈s1, . . . , sn〉 in I .

Usually the model I of the previous definition is a linear order or a tree, but ingeneral it can be any model.

Definition 3.2. Let θ be a linear order. Let Kωtr (θ) be the class of models that are

isomorphic to a model of the form

I = (M,�, Pα, <, H)α≤ω

where M ⊆ ≤ωθ and the following conditions hold.


(i) M is closed under initial segments.(ii) � denotes the initial segment relation.

(iii) Pα = {η ∈ M | dom(η) = α}.(iv) < denotes the lexicographic ordering on M .(v) H(η, ν) is the maximal common initial segment of η and ν.

The structures in Kωtr (θ) are called ordered trees. Let

Kωtr =

⋃{Kω

tr (θ) | θ is a linear order}.Definition 3.3. (i) Let τ be a vocabulary, and let τ1 ⊇ τ be a vocabulary con-

taining Skolem functions for the language Lωω(τ). A τ1-structure M is anEhrenfeucht-Mostowski model if there are a model I , sequences as for s ∈ I ,and a function � such that the following hold:(a) The function � maps the quantifier free type of s = 〈s1, . . . , sn〉 in I to

the quantifier free type of as1 � . . . � asn in M.(b) The collection 〈as | s ∈ I 〉 is quantifier free indiscernible in M.(c) The structure M is generated by 〈as | s ∈ I 〉.

(ii) The model I is called the index model of the Ehrenfeucht-Mostowski modelM and the collection 〈as | s ∈ I 〉 that generates M is called the skeleton (ofM).

(iii) The function � is called a template and we say it is proper for I if there isan Ehrenfeucht-Mostowski model M with I as the index model. We say � isproper for a class K of models if it is proper for every model I ∈ K .

Remark 3.4. If M, as , s ∈ I , and M′, a′s , s ∈ I , are Ehrenfeucht-Mostowski modelsconstructed using a template � (which is proper for I ), then the mapping whichtakes a = µ(as) in M to a′ = µ(a′s ) in M′ (for all τ1-terms µ and sequencess ⊆ I ) is an isomorphism between M and M′. Since the Ehrenfeucht-Mostowskimodel is unique up to isomorphism, it is denoted by EM1(I, �). By EM(I, �) wedenote EM1(I, �)�τ (the restriction of the structure EM1(I, �) to the vocabularyτ ). ��Lemma 3.5. Suppose that a template� is proper for I0 and I1. LetA = EM1(I0, �)

and B = EM1(I1, �), and let 〈as | s ∈ I0〉 and 〈bs | s ∈ I1〉 be the skeletons ofthe models. Suppose further that |I0| = |I1| = λ.

(i) Suppose f ∈ Part(I0, I1). Let f ′ be defined as follows: For every τ1-term µ

and s1, . . . , sn ∈ dom(f ),

f ′(µ(as1 , . . . , asn)) = µ(bf (s1), . . . , bf (sn)).

Then f ′ is an isomorphism from hull(A, {as | s ∈ dom(f )}) onto hull

(B, {bs |

s ∈ ran(f )}).(ii) Suppose that U is a λ-tree, and κ is an infinite cardinal. If furthermore I0 ≡κ

U

I1, then A ≡κU B.

Proof. (i) Follows from the fact that the skeletons are quantifier free indiscerniblein the models and that the models are generated by the same template �.

(ii) Follows easily from (i). ��


Theorem 3.6 ([17, Theorem 1.3]). Suppose T be a complete first order theory in acountable vocabulary τ . Suppose further that T is unsuperstable and the first-orderformulas ϕn(x, yn), n < ω, witness this. Then there are a countable Skolemizedtheory T1 ⊇ T in vocabulary τ1 and a template � proper for Kω

tr such that forevery I ∈ Kω

tr and for all η ∈ P Iω , ν ∈ P I

n ,

EM1(I, �) |= T1,

EM1(I, �) |= ϕn(aη, aν) iff ν � η

where 〈aη | η ∈ I 〉 is the skeleton. Furthermore, EM1(I, �) is of cardinality|I | + ω.

4. Guessing clubs

The guessing club principle was discovered by Shelah. In this section we present avariation of guessing club principles.

Definition 4.1. Suppose κ < λ are regular cardinals and S ⊆ λ. A sequence〈aα | α ∈ lim(S) ∩ Sλ

κ 〉 is a κ-club-guessing sequence for S if the followingconditions hold:

(i) Each aα = 〈aαi | i < κ〉 is an increasing sequence of elements of S cofinal in

α.(ii) For every club subset C of λ it holds that

{α ∈ lim(S) ∩ Sλκ | aα ⊆ C} �= ∅.

Remark 4.2. We get an equivalent definition for a κ-club-guessing sequence for S

if we replace condition (ii) of Definition 4.1 by the following: For every club subsetC of λ the set

{α ∈ lim(S) ∩ Sλκ | aα ⊆ C}

is stationary in λ.

Theorem 4.3. (i) If κ , λ are regular cardinals and κ+ < λ, then the existence ofa κ-club-guessing sequence for λ is provable in ZFC [13, Claim 7.8A].

(ii) Suppose κ < µ < λ are regular cardinals. Then the existence of a κ-club-guessing sequence for Sλ≥µ is provable in ZFC [13, Claim 7.8B]. ��

The references in the theorem are to a manuscript of [13] of September 1991.Additional material can be found in [18].

Lemma 4.4. Suppose µ is an uncountable regular cardinal, κ < µ is a regularcardinal, λ = µ+, and S ⊆ Sλ

µ. Then there is a κ-club-guessing sequence for S or

Sλµ\S.


Proof. Let S′ = Sλµ\S. By (ii) in Theorem 4.3 and the fact that λ = µ+, there is a

κ-club-guessing sequence 〈aα | α ∈ lim(Sλµ)∩ Sλ

κ 〉 for Sλµ. For each α ∈ lim(S)∩

Sλκ , define

bα ={

aα ∩ S if sup(aα ∩ S) = α,

bα otherwise

where bα ⊆ S is some unbounded subset of α of order type κ . Similarly, for eachα ∈ lim(S′) ∩ Sλ

κ , define

cα ={

aα ∩ S′ if sup(aα ∩ S ′) = α,

cα otherwise

where cα ⊆ S′ is some unbounded subset of α of order type κ .We claim that at least one of

b = 〈bα | α ∈ lim(S) ∩ Sλκ 〉,

c = 〈cα | α ∈ lim(S′) ∩ Sλκ 〉

is a κ-club-guessing sequence as required. Towards a contradiction assume thatthere are sets C, C′ ⊆ λ club in λ that avoid the sequences b and c, respectively.Since the set D = C ∩ C′ is club in λ, there is α ∈ lim(Sλ

µ) ∩ Sλκ with aα ⊆ D. It

follows from aα ⊆ S∪S′ that sup(aα∩S) = α or sup(aα∩S′) = α. Without loss ofgenerality we may assume that sup(aα∩S) = α. Hence α ∈ lim(S)∩Sλ

κ . Thereforebα = aα ∩ S and it follows that bα ⊆ D ⊆ C contradicting the assumption that C

avoids b. (4.4)��Corollary 4.5. Suppose µ is an uncountable regular cardinal, κ < µ is aregular cardinal, λ = µ+, and S ⊆ Sλ

µ. If S′ ⊆ S is µ-club in λ, then thereis a κ-club-guessing sequence for S.

Definition 4.6. Suppose κ < λ are regular cardinals, and S ⊆ λ. The partialorder Pκ(S) is defined so that its universe consists of the functions p that satisfythe following conditions.

(i) There is β < λ such that dom(p) = lim(S) ∩ Sλκ ∩ β.

(ii) For all α ∈ dom(p), p(α) ∈ κS is increasing and cofinal in α.

The functions are ordered by end-extension.

Lemma 4.7. Suppose κ < λ are regular cardinals, and S ⊆ λ is stationary. ThenPκ(S) is λ-closed and in every Pκ(S)-generic extension there is a κ-club-guessingsequence for S.

Proof. Let P denote Pκ(S). Clearly, P is λ-closed. Hence S remains stationary inP -generic extensions.

Let G be a P -generic set. We claim that g = ⋃G is a κ-club-guessing

sequence for S in V [G]. Towards a contradiction assume that it is not. Then thereare a P -name C and a condition p ∈ G such that

p �P C is a club ∧ ∀α ∈ lim(S) ∩ Sλκ (g(α) �⊆ C).


Recursively define a decreasing sequence 〈pi | i < κ〉 of conditions and anincreasing sequence 〈βi | i < κ〉 of ordinals such that

p0 = p,

βi > sup(dom(pi)),

pi+1 � βi ∈ S ∩ C.

This is possible as P is λ-closed and p forces “C is a club”. Let q =⋃i<κ pi and

β =⋃i<κ βi . Since P is λ-closed and κ < λ, q is in P . By the construction of the

sequences, β /∈ dom(q). Hence q ′ = q ∪ {〈β, 〈βi | i < κ〉〉} is also in P . Now

q ′ � β ∈ lim(S) ∩ Sλκ ∧ g(β) ⊆ C ∧ g(β) �⊆ C

which is absurd. Hence the claim follows. ��

5. A consistency result on unsuperstable theories

In this section we show the following: It is consistent relative to the existence of aMahlo cardinal that every complete unsuperstable theory T in a countable vocabu-lary τ has two non-isomorphic models A and B of cardinality λ such that A ≡λ

α Bfor every α < λ. In fact, we will show that the models are much more equivalentthan what is stated above. The following lemma is crucial in this context. It showsthat if player ∃ has a winning strategy in GCκ(Sκ+

κ \S, κ+) then she has winningstrategies in much longer games, too.

Lemma 5.1. Suppose κ is a regular cardinal, λ = κ+ and S ⊆ λ.

(i) If player ∃ has a winning strategy in GCκ(S, λ), then she has a winningstrategy in GCκ

α(S, λ) for every α < λ.(ii) Suppose that U0, U1 and Ui , i ∈ I , are trees of height at most λ such that

player ∃ has a winning strategy in GCκU (S, λ) where U is any of the trees U0,

U1, Ui . Then she has winning strategies in GCκU0+U1

(S, λ), GCκU0·U1

(S, λ),GCκ⊕

i∈I Ui(S, λ) and GCκ⊗

i∈I Ui(S, λ).

Proof. (i) Let w be a winning strategy for player ∃ in GCκ(S, λ). The claim isproved by induction on α. Suppose that for all β < α the claim holds. The claimis clear for successor ordinals. So, assume that α is a limit ordinal. Let µ = cf(α),and let 〈αi | i < µ〉 be an increasing sequence cofinal in α. For i < µ, let wi bea winning strategy for player ∃ in GCκ

αi(S, λ) and βi = �j<i(αj + 1). Note that

the sequence 〈βi | i < µ〉 is continuous. Suppose ξ < α. Let 〈xζ | ζ ≤ ξ〉 bethe choices of player ∀ thus far. Let k = sup{i < µ | βi ≤ ξ}. The choice yξ ofplayer ∃ on round ξ is defined as follows:

yξ ={

w(xβ0 , xβ1 , . . . , xβk) if βk = ξ,

wk(xβk+1, . . . , xξ ) otherwise.


By using the strategy described above player ∃wins the game GCκα(S, λ). Towards

a contradiction suppose that she does not win. Then there is β ≤ α of cofinality κ

such that

sup{xξ | ξ < β} /∈ S.

If β = α, we have

sup{xξ | ξ < β} = sup{xβi| i < µ}.

Since the pair 〈〈xβi| i < µ〉, 〈yβi

| i < µ〉〉 is a play in which player ∀ playsagainst the winning strategy w, we have

sup{xβi| i < µ} ∈ S.

This is a contradiction. So β < α and there is i < µ with βi ≤ β < βi+1. Sincecf(α) = µ ≤ κ , cf(i) < κ . Hence cf(βi) < κ (if i is a successor ordinal, then βi

is a successor ordinal) and it follows that βi < β. Since yξ = wi(xβi+1, . . . , xξ )

for βi < ξ < βi+1 and wi is a winning strategy for player ∃, we have

sup{xξ | ξ < β} = sup{xξ | βi < ξ < β} ∈ S.

Again we have a contradiction. Thus player ∃ wins.(ii) The cases GCκ⊕

i∈I Ui(S, λ) and GCκ⊗

i∈I Ui(S, λ) are obvious. As to the case

GCκU0+U1

(S, λ), first use a winning strategy in GCκU0

(S, λ) and then a winningstrategy in GCκ

U1(S, λ). The case GCκ

U0·U1(S, λ) is proved similarly to (i). (5.1)��

Definition 5.2. For a regular cardinal κ and a cardinal λ, let Uκ,λ be the least setof (κ, λ)-trees that contains the ordinals α, α < κ , and is closed under the treeoperations +, ·, ⊕ and ⊗. Define

Uκ =⋃

λ

Uκ,λ.

Corollary 5.3. Suppose that S ⊆ Sκ+κ is a stationary subset of κ+ and player ∃ has

a winning strategy in GCκ(S, κ+). Then she has a winning strategy in GCκU (S, κ+)

for all U ∈ Uκ+ .

The construction is a modification of the one in [6], but for the reader’s con-venience and because we will use the construction again in Section 7 we give theconstruction below.

Lemma 5.4 ([6, Lemma 2.12]). Assume κ is a regular cardinal and λ = κ+. Thenthere are a linear order θ of cardinality λ, a bijection h : θ → λ× θ and automor-phisms gα : θ → θ for each α < λ such that the following conditions hold.

(i) If gα(x) = y, then x �= y, and either h(x) = 〈α, y〉 or h(y) = 〈α, x〉, but notboth.

(ii) If gα(x) = gβ(x), then α = β.(iii) If h(x) = 〈α, y〉, then gα(x) = y or gα(y) = x.


Assumption 5.5. From now on in the section we assume κ > ω1 is a regularcardinal, κω = κ , and λ = κ+, unless explicitly stated otherwise.

By θ0 we denote the linear order λ, by θ1 the linear order given by Lemma 5.4,and θ2 the linear order θκ·ω · λ where θκ·ω is <ω(κ · ω) ordered lexicographically.Let h and gα , α < λ, be as in Lemma 5.4. We may assume that the linear ordersθ0, θ1 and θ2 are pairwise disjoint. Hence, let

θ = θ0 ∪ θ1 ∪ θ2

and order it so that the elements of θ0 come first, then the elements of θ1 and lastlythe elements of θ2.

Definition 5.6 ([6, Section 3]).

(i) For i < ω we say that i is of type k < 3 if i ≡ k mod 3. The type of i isdenoted by type(i).

(ii) For n = 0, 1, define ordered trees J−n , J+n ∈ Kωtr (θ) as follows:

J−n = {ρ ∈ <ωθ | ρ �= 〈〉 ∧ ρ(0) = n ∧ ∀i ∈ dom(ρ)(ρ(i) ∈ θtype(i))},J+n = {ρ ∈ ≤ωθ | ρ �= 〈〉 ∧ ρ(0) = n ∧ ∀i ∈ dom(ρ)(ρ(i) ∈ θtype(i))}.

(iii) Let

f : D0 → λ\{0}be a bijection where

D0 = {〈η, ξ〉 ∈ J−0 × J−1 | dom(η) = dom(ξ) is of type 1}.Usually we write f (η, ξ) instead of f (〈η, ξ〉). For every 〈η, ξ〉 ∈ D0 define afunction gη,ξ : succ(η)→ succ(ξ) by

gη,ξ (η � 〈x〉) = ξ � 〈gf (η,ξ)(x)〉.(iv) Let

e : θ1 → λ

be a bijection with e(z) = 0. For every η ∈ J+n and i ∈ dom(η), let

s(η, i) =

η(i) if type(i) = 0,

e(η(i)) if type(i) = 1,

2nd(η(i)) if type(i) = 2.

For every β < λ and J ⊆ J+n , let

J+n (β) = {ρ ∈ J+n | ∀i ∈ dom(ρ)(s(ρ, i) < β)},J (β) = J ∩ J+n (β).


Lemma 5.7 ([6, Lemma 3.1]). There is a function d : θ1 → J−0 × J−1 such thatfor every η ∈ J−0 and ξ ∈ J−1 the following holds: If dom(η) = dom(ξ) = j + 1is of type 2 and gη�j,ξ�j (η) = ξ , then either

d(η(j)) = ξ or d(ξ(j)) = η

but not both.

Definition 5.8 ([6, Section 3]).

(i) For all η ∈ J+0 and ξ ∈ J+1 define a relation R0 by ηR0ξ if and only if(a) for all non-zero j ∈ dom(η) ∩ dom(ξ) of type 0, η(j) = ξ(j),(b) for all j ∈ dom(η) ∩ dom(ξ) of type 1,

ξ�(j + 1) = gη�j,ξ�j (η�(j + 1)).

(ii) For η ∈ J−0 and ξ ∈ J−1 such that dom(η) = dom(ξ) = j + 1 is of type 2 andηR0ξ we define a direction relation � as follows:

η � ξ iff d(η(j)) = ξ,

ξ � η iff d(ξ(j)) = η.

Note that, by Lemma 5.7, the direction is defined for every ηR0ξ with dom(η) =dom(ξ) of type 2, and we have either η � ξ or ξ � η, but not both.

Definition 5.9. For any subset S of λ define:

(i) For all η ∈ J+0 and ξ ∈ J+1 with dom(η) = dom(ξ) = ω, define a relation RS1

by ηRS1 ξ if and only if the following conditions hold.

(a) ηR0ξ .(b) For all i < ω of type 2, it holds that

η�i � ξ�i → s(η, i) ≤ s(ξ, i),

ξ�i � η�i → s(ξ, i) ≤ s(η, i).

(c) For all i < ω, if i is of type 2, ξ�i � η�i and s(η, i) ∈ S, then s(ξ, i) <

s(η, i).(ii) Let RS be the symmetric closure of RS

1 .(iii) A pair of ordered trees J0 and J1 is said to be S-obedient if and only if the

following two conditions hold:(a) J−n ⊆ Jn ⊆ J+n , n = 0, 1,(b) if η ∈ J+0 , ξ ∈ J+1 and ηRS

1 ξ , then η ∈ J0 ↔ ξ ∈ J1.

Definition 5.10. (i) Suppose J0 and J1 form an S-obedient pair. Define a set offunctions GJ0,J1 such that f ∈ GJ0,J1 if and only if the following conditionshold.(a) f ∈ Part(J0, J1).(b) dom(f ) and ran(f ) are closed under initial segments, and f is of cardi-

nality at most κ .(c) For every η ∈ dom(f ), ηR0f (η) (c.f 5.9(i-a)).


(d) For every η ∈ dom(f ) and i ∈ dom(η) of type 2, it holds that

η�i � ξ�i → s(η, i) ≤ s(ξ, i),

ξ�i � η�i → s(ξ, i) ≤ s(η, i)

where ξ = f (η) (c.f 5.9(i-b)).(e) For every η ∈ dom(g) and i ∈ dom(η) of type 2, it holds that

(ξ�i � η�i ∧ s(η, i) ∈ S)→ s(ξ, i) < s(η, i)

where ξ = f (η) (c.f 5.9(i-c)).(f) For every η ∈ dom(f ) with dom(η) of type 2 there is an ordinal γ such

that γ = 0 or γ ∈ λ\S, and moreover

γ = {α < λ | ∃x ∈ θκ·ω

(η � 〈〈x, α〉〉 ∈ dom(f )

)}

= {α < λ | ∀x ∈ θκ·ω

(η � 〈〈x, α〉〉 ∈ dom(f )

)}

= {α < λ | ∃x ∈ θκ·ω

(f (η) � 〈〈x, α〉〉 ∈ ran(f )

)}

= {α < λ | ∀x ∈ θκ·ω

(f (η) � 〈〈x, α〉〉 ∈ ran(f )

)}.

We denote this ordinal γ by o(f, η).(ii) Define G

J0,J11 to be the set

{f ∈ GJ0,J1 | ∃γ < λ∀η ∈ dom(f )

(type(dom(η)) = 2 → o(f, η) ∈ {0, γ })}.

The ordinal γ in the formula above related to f is denoted by o(f ).(iii) For f ∈ GJ0,J1 define f to be a function such that

dom(f ) = dom(f ) ∪ {η ∈ J0 | dom(η) = ω ∧ ∀i < δ(η�i ∈ dom(f ))}and for each η ∈ dom(f )\dom(f ),

f (η) =⋃

i<dom(η)

f (η�i).

Lemma 5.11. Suppose S ⊆ Sλκ , κω = κ , and J0 and J1 form an S-obedient pair.

(i) If f ∈ GJ0,J1 and A ⊆ J0 ∪ J1 is of cardinality at most κ , then there isf ′ ∈ G

J0,J11 such that f ⊆ f ′ and A ⊆ dom(f ′) ∪ ran(f ′).

(ii) Suppose δ < λ and 〈fξ | ξ < δ〉 is an ascending chain of functions in GJ0,J11

such that o(fζ ) < o(fξ ) whenever ζ < ξ . If furthermore sup{o(fξ ) | ξ <

δ} /∈ S, then⋃

ξ<δ fξ ∈ GJ0,J11 .

Proof. This can be proved as [6, Lemma 3.11]. ��Lemma 5.12. Suppose that S ⊆ Sλ

κ , the trees J0 and J1 form an S-obedient pair,and U is a λ-tree such that there is a winning strategy for player ∃ in GCκ

U (Sλκ \S, λ).

Then

J0 ≡λU J1.


Proof. The proof is analogous to the proof of [6, Theorem 3.13]. The winningstrategy for player ∃ in GCκ

U (Sλκ \S, λ) is used to ensure that at limit steps con-

dition 5.10(i-f) holds, i.e., we make sure that for a limit ordinal δ it holds thatsup{o(fξ ) | ξ < δ} ∈ λ\S where the functions fξ are partial isomorphisms guid-ing the choices of player ∃. ��

Note that in the following definition the objects D0, e, f and z are as in Defi-nition 5.6.

Definition 5.13. Let S ⊆ Sλκ be stationary in λ and a = 〈aα | α ∈ lim(S) ∩ Sλ

ω〉an ω-club-guessing sequence for S.

(i) For every 〈η, ξ〉 ∈ D0, let

C(η, ξ) = {α < λ | x ∈ θ1

(e(x) < α ↔ e(gf (η,ξ)(x)) < α

)}.

For every β < λ, let

σ(β) = min⋂ {

C(η, ξ) | 〈η, ξ〉 ∈ D0 ∩ (J+0 (β)× J+1 (β))}.

(ii) An ordinal α < λ is active if α ∈ lim(S)∩Sλω, and there are a strictly increas-

ing subsequence b = 〈bi | i < ω〉 of aα and η ∈ J+0 with dom(η) = ω suchthat the following conditions hold:(a) For every i < ω, bi+1 > σ(bi).(b) For every non-zero i < ω of type 0 or 2, s(η, i) = bi .(c) For every i < ω of type 1, η(i) = z.For an active ordinal α, let ηα and bα witness the activeness of α.

(iii) By recursion on α < λ we define Iα0 and Iα

1 as follows:(a) I 0

0 = J−0 and I 01 = J−1 .

(b) If α is active, define Iα0 and Iα

1 to be the least sets such that

{ηα} ∪⋃

β<α

Iβ0 ⊆ Iα

0 and⋃

β<α

Iβ1 ⊆ Iα

1 ,

Iα0 ∪ Iα

1 is closed under RS.

(c) If α is not active, put Iαn =

⋃β<α I

βn for n = 0, 1.

(iv) Define

J Sn =

⋃

α<λ

Iαn , n = 0, 1.

Note that the sets C(η, ξ) in 5.13(i) are club subsets of λ. Since we assumeκω = κ , the sets J+0 (β) and J+1 (β) are of cardinality κ for every β < λ. Thus, inthe definition of σ(β) the intersection is non-empty.

Note also that by the construction of J S0 and J S

1 , they form an S-obedient pair.The following lemma is essentially Lemma 3.18 in [6].


Lemma 5.14. Suppose T is a complete unsuperstable theory in a countable vocab-ulary τ , κω = κ , S ⊆ Sλ

κ is a stationary subset of λ, there is an ω-club-guessingsequence for S and � is a template proper for Kω

tr given by Theorem 3.6. Then

EM(J S0 , �) �∼= EM(J S

1 , �).

Corollary 5.15. Let T be a complete unsuperstable theory in a countable vocabu-lary τ and � a template proper for Kω

tr given by Theorem 3.6. Suppose

κω = κ ,S ⊆ Sλ

κ is a stationary subset of λ,U is a λ-tree such that player ∃ has a winning strategy in GCκ

U (Sλκ \S, λ),

there is an ω-club-guessing sequence for S.

Then

EM(J S0 , �) ≡λ

U EM(J S1 , �) and EM(J S

0 , �) �∼= EM(J S1 , �).

Proof. The corollary follows from (ii) of Lemma 3.5, Lemma 5.12, andLemma 5.14. ��Theorem 5.16. Suppose κ is a regular cardinal, there is an uncountable cardinalθ < κ with θ<θ = θ and λ > κ is a Mahlo cardinal. Then there is a partial order P

such that in every P -generic extension the following hold.

(i) κ is still a regular cardinal, all cardinals less than κ are preserved, andκ+ = λ.

(ii) For every theory T such as in Corollary 5.15 there are non-isomorphic modelsA and B of cardinality λ such that for every tree U ∈ Uλ,

A ≡λU B.

Proof. We will show that there is a forcing extension in which κω = κ , λ = κ+,and in which there is a set S ⊆ Sλ

κ stationary in λ such that there are

a winning strategy for player ∃ in GCκ(Sλκ \S, λ),

an ω-club-guessing sequence for S.

Then, by Corollary 5.3 and Corollary 5.15, the theorem follows.

Let S = (Sλin)

V \κ . Force with Mitchell’s partial order R(θ, κ, λ) (see Defini-tion 6.6). By Lemma 6.7, in the generic extension we have κ+ = λ, κω = κ , theset S is a subset of Sλ

κ , and player ∃ has a winning strategy w in GCκ(Sλκ \S, λ). We

still need to have an ω-club-guessing sequence for S. We can have this by forcingwith Pω(S) of Definition 4.6.

Since Pω(S) is λ-closed, S is preserved as a κ-stationary subset of λ and w willstill be a winning strategy for player ∃ in GCκ(Sλ

κ \S, λ). ��By the theorem above, we can have a universe in which every countable unsu-

perstable theory T has non-isomorphic models A and B of cardinality ω3 such thatfor every U ∈ Uω3 , A ≡ω3

U B.


6. Equiconsistency results

In this section we prove the following theorem.

Theorem 6.1. The following are equiconsistent:

(i) There is a Mahlo cardinal.(ii) There is a stationary S ⊆ S

ω2ω1 such that player ∃ has a winning strategy in the

club game GCω1(Sω2ω1 \S, ω2).

(iii) There are (ω3, ω + 2)-trees I0 and I1 such that I0 �∼= I1 and ∃ has a winningstrategy in the Ehrenfeucht-Fraısse game EFω2

U (I0, I1) for all U ∈ Uω2 .(iv) There are (ω3, ω + 2)-trees I0 and I1 such that I0 �∼= I1 and ∃ has a winning

strategy in the Ehrenfeucht-Fraısse game EFω2ω1·ω+1(I0, I1).

(v) I [ω2] is proper.

Furthermore, we can additionally require the condition ω<ω22 > ω2 in any of the

cases above.

One should note that in the theorem above we can have ω<ω22 > ω2 but in

Theorem 5.16 we were not able to have the analogous property ω<ω33 > ω3. This is

because in the last part of the proof of Theorem 5.16 forcing with the partial orderPω(Sλ

in), which adds the required ω-club-guessing sequence, collapses ω<ω33 to ω3.

Another fact worth noting is that 2ω = ω2 holds in the generic extension whichwe obtain when starting from a Mahlo cardinal. Hence the trees I0 and I1 of thetheorem will be of cardinality ω2, as desired.

We will actually prove more than what is stated in the above theorem, mostof the relative consistency implications are really implications, i.e., instead of justhaving Con(A) → Con(B) we have A → B. The following table summarizes theresults.

Implication LemmaCon(i)→ Con(ii) Lemma 6.7(ii)→ (iii) Lemma 6.13(iii)→ (iv) Immediate(iv)→ (v) Lemma 6.4Con(v)→ Con(i) Lemma 6.5(ii)→ (v) Lemma 6.3

We stated the theorem using ω2, but there is nothing special with ω2 except that itis the least cardinal for which the implication sequence from (ii) to (v) can hold.We state the lemmas which show the individual implications more generally.

Remark 6.2. If ωω2 = ω2, then (v)→ (ii), and hence

(ii)↔ (iii)↔ (iv)↔ (v).

��Lemma 6.3 ([4, Lemma 2.2]). Let κ be a regular cardinal and λ = κ+. Supposethat S ⊆ Sλ

κ is a stationary subset of λ and player ∃ has a winning strategy inGCκ(Sλ

κ \S, λ). Then I [λ] is proper.


Lemma 6.4 ([5, Theorem 4.4]). Let κ be a regular cardinal and λ = κ+. Supposethat I0 and I1 are (λ+, ω+2)-trees such that I0 �∼= I1 and ∃ has a winning strategyin EFλ

κ·ω+1(I0, I1). Then I [λ] is proper.

Lemma 6.5. Let λ be a regular cardinal. Suppose that I [λ] is proper. Then λ is aMahlo cardinal in L.

Proof. Towards a contradiction suppose that λ is not a Mahlo cardinal in L. ThenI [λ] is improper in L. Let C ⊆ λ and a = 〈aξ | ξ < λ〉 witness that. Suppose thatδ ∈ C. Then in L there is e ⊆ δ unbounded in δ with order type less than δ suchthat for every α < δ there is ξ < δ with e ∩ α = aξ . Since everything above isabsolute for transitive models of ZFC, the objects witnessing the improperness ofI [λ] in L witness the improperness of I [λ] in V contradicting the assumption thatI [λ] is proper. Hence λ is a Mahlo cardinal in L. ��

6.1. Proof of Con(i)→ Con(ii)

Let P be a separative partial order and A a subset of P . By B(P, A) we denote thecollection of regular open subsets of P generated by subsets of A. By B(P ) wedenote B(P, P ), i.e., the Boolean completion of P .

A subset A of P is called pre-dense if for every p ∈ P there is q ∈ A which iscompatible with p.

Let θ be a cardinal and µ an ordinal. By Fn(µ, 2, θ) we denote the set of partialfunctions from µ to 2 that are of cardinality < θ ordered by reverse inclusion, i.e.,the Cohen forcing that adjoins |µ| new subsets of θ .

Definition 6.6 ([11, p. 26]). Let θ < κ < λ be regular cardinals. Mitchell’spartial order related to the cardinal parameters θ ,κ and λ, R(θ, κ, λ), is definedas follows: For γ ≤ λ, let Pγ denote Fn(γ, 2, θ). We write P for Pλ. Let Q be thecollection of functions f which satisfy

(i) dom(f ) ⊆ λ, and ran(f ) ⊆ B(P ),(ii) |f | < κ ,

(iii) ∀γ ∈ dom(f )(f (γ ) ∈ B(P, Pγ+θ )

).

The partial order R(θ, κ, λ) is defined to be the two stage iterated forcing

R(θ, κ, λ) = P ∗ Q

where the order of 〈p, f 〉, 〈q, g〉 ∈ R(θ, κ, λ) is defined as follows: 〈p, f 〉 ≤ 〈q, g〉if and only if

(iv) dom(g) ⊆ dom(f ),(v) p �P ∀γ ∈ dom(g)(g(γ ) ∩ G = ∅ ↔ f (γ ) ∩ G = ∅).

Lemma 6.7. Suppose θ < κ are regular cardinals with 2<θ = θ , λ > κ is a Mahlo

cardinal and µ ≥ λ a cardinal with cf(µ) ≥ λ. Let S = (Sλin)

V \κ . Then in everyFn(µ, 2, κ)× R(θ, κ, λ)-generic extension the following hold.


(i) All cardinals ≤ κ and cardinals ≥ λ are preserved.(ii) κ+ = λ, 2<θ = θ , 2θ = λ and 2κ = µ.

(iii) The set S is κ-stationary and there is a winning strategy for player ∃ inGCκ(Sλ

κ \S, λ).

Proof. Suppose that µ ≥ λ. We will not give a detailed proof of the lemma but justtry to give the main points from which the lemma follows. The key observationsare as follows. Let ν be an inaccessible cardinal ≤ λ and > κ .

(1.1) Since Fn(µ, 2, κ) is κ-closed, it preserves cardinals up to κ . As ν is inac-cessible and it is greater than κ , we have (2<κ)+ < ν. Hence Fn(µ, 2, κ)

preserves cardinals ≥ ν.(1.2) The inaccessibility of ν is needed only in the proof that R(θ, κ, ν) has the

ν-c.c. To prove the other key properties of Mitchell’s forcing it suffices toassume 2<θ = θ .

(1.3) Fn(µ, 2, κ)× R(θ, κ, ν) has the ν-c.c.(1.4)

(〈R(θ, κ, ν),≤〉)V = (〈R(θ, κ, ν),≤〉)V [G].

It follows from these observations that all cardinals strictly between κ and λ arecollapsed to κ and all other cardinals are preserved. The observations also implythe following property of Mitchell’s forcing. Let G be an Fn(µ, 2, κ)-generic setand K an R(θ, κ, λ)-generic set over V [G]. Then

(2.1) for every ρ ≤ ν with ∀α < ρ(α + θ ≤ ρ) and γ with cf(γ ) > θ , ift : γ → V ∈ V [G][Kν] and t�α ∈ V [G][Kρ] for every α < γ , thent ∈ V [G][Kρ].

This property is needed in the proof that player ∃ has a winning strategy inGCκ(Sλ

κ \S, λ).Let R(ν) denote R(θ, κ, ν) and Pν denote Fn(ν, 2, θ).

Claim 6.8. Suppose that ν ≤ λ is inaccessible > κ . Then Fn(µ, 2, κ)× R(ν) hasthe ν-c.c.

Proof. Let A = {〈pξ , qξ , fξ 〉 | ξ < ν} be a subset of Fn(µ, 2, κ) × R(ν). Sinceν is inaccessible and ν > κ > θ , there is Y1 ⊆ ν of cardinality ν such that for allξ, ζ ∈ Y1, pξ ‖ pζ and qξ ‖ qζ .

Let D = {dom(fξ ) | ξ ∈ Y1}. Since every acceptable function f is of cardi-nality less than κ and ν is inaccessible, there is, by the �-system lemma, X ⊆ Y1of cardinality ν such that the set {dom(fξ ) | ξ ∈ X} form a �-system with root r .Let α = sup(r). Then for every ξ ∈ X,

fξ �r ∈ rB(Pν, Pα+θ

).

Since α + θ < ν and ν is inaccessible, |Pα+θ | < ν. Hence

|rB(Pν, Pα+θ

)| < ν.


Therefore, by the pigeon hole principle, there is Y2 ⊆ X of cardinality ν suchthat for every ξ, ζ ∈ Y2, fξ �r = fζ �r . Hence, for every ξ, ζ ∈ Y2, the con-dition 〈pξ ∪ pζ , qξ ∪ qζ , fξ ∪ fζ 〉 is an extension of 〈pξ , qξ , fξ 〉 and 〈pζ , qζ ,

fζ 〉. (6.8) ��Let ν ≤ λ be an inaccessible cardinal and G a Fn(µ, 2, κ)-generic set. It

follows from the previous claim and the fact that Fn(µ, 2, κ) has the ν-c.c., that(〈R(ν),≤〉)V has the ν-c.c. in V [G].

Claim 6.9. Suppose that ν ≤ λ is inaccessible > κ . Let G be a Fn(µ, 2, κ)-genericset. Then

(〈R(ν),≤〉)V = (〈R(ν),≤〉)V [G].

Proof. Since Fn(µ, 2, κ) is κ-closed and 2<θ = θ , Pν has the θ+-c.c. in V [G] andfor all α ≤ ν,

Pα = (Pα)V [G],

B(Pν, Pα

) = (B(Pν, Pα)

)V [G].

Towards a contradiction suppose that 〈R(ν),≤〉 �= (〈R(ν),≤〉)V [G]. First we notethat

R(ν) ⊆ (R(ν))V [G].

So we have two possibilities: There is 〈p, f 〉 ∈ (R(ν))V [G]\R(ν), or there are〈p1, f1〉, 〈p2, f2〉 ∈ R(ν) with

〈p1, f1〉 ≤R(ν) 〈p2, f2〉 ↔ (〈p1, f1〉 �≤R(ν) 〈p2, f2〉)V [G].

The first case is impossible, since there are no new subsets of cardinality less than κ

in V [G]. For the latter possibility we note that, by the definition of 〈R(ν),≤〉,〈p1, f1〉 ≤R(ν) 〈p2, f2〉 if and only if

p1 ≤ p2 ∧∀α ∈ dom(f2)∀q ≤ p1(q ∈ f1(α)→ f2(α) is pre-dense below q

∧ q ∈ f2(α)→ f1(α) is pre-dense below q).

But this is absolute for V and V [G], as R(ν) = (R(ν))V [G]. Hence the latteralternative is impossible, too, and the claim follows. ��

By the claims above, we have in V [G] that (R(ν))V [G] has the ν-c.c.Since λ is a Mahlo cardinal in V , the set S is bistationary in V . By Claim 6.8,

Fn(µ, 2, κ)× R(λ) has the λ-c.c. Hence S is bistationary in V [G×K].It remains to show that player ∃ has a winning strategy in in GCκ(Sλ

κ \S, λ).Work in V [G×K]. A winning strategy is as follows: Suppose α < κ and player ∀has chosen the ordinals {xζ | ζ ≤ α}. If α = 0, then player ∃ chooses y0 < λ withy0 > x0 and y0 > (2κ)V . Otherwise she chooses an ordinal yα < λ such that


(3.1) yα > xα ,(3.2) for all γ < yα it holds that γ + θ ≤ yα ,(3.3) {xζ | ζ ≤ α} ∈ V [G×Kyα ].

Towards a contradiction suppose that player ∀ wins by choosing the ordinals {xα |α < κ}. Let ρ = sup{xα | α < κ}. As player ∀ wins, ρ ∈ S and it follows that forall α < ρ we have α + θ ≤ ρ. By (2.1),

{xα | α < κ} ∈ V [G×Kρ].

Hence

V [G×Kρ] |= cf(ρ) = κ.

But this contradicts the fact that ρ being inaccessible in V , is the successor of κ inV [G×Kρ]. (6.7)��

In [11, Section 6] Mitchell uses a similar modification as the one we introducedabove.

6.2. Proof of (ii)→ (iii)

Our approach in the section is the same as in Section 5. We take a construction, thistime from [5], and by modifying it in the similar vein as in Section 5 we get thedesired result. In this section we assume that κ is an uncountable regular cardinaland λ = κ+.

Lemma 6.10 ([5, Lemma 3.1]). Suppose S ⊆ Sλκ is stationary. Then there is a

partition {Sα | α < λ} of S ∪ Sλ<κ such that for all α < λ,

(i) Sα ∩ Sλκ is a stationary subset of λ, and

(ii) if δ ∈ Sα and cf(δ) = κ , then δ = sup(δ ∩ Sα ∩ Sλ<κ).

Suppose S ⊆ Sλκ is stationary. Using a partition S of S given by the above lemma

relations R0 and RS1 can be defined as in [5]. We do not repeat the definitions but

ask the interested reader to refer to [5].The following lemma can be proved in a similar fashion as [5, Theorem 3.3].

Lemma 6.11. Suppose that S ⊆ Sλκ is stationary such that player ∃ has a winning

strategy in GCκU (Sλ

κ \S, λ), and I0 and I1 are trees such that

(i) I−n ⊆ In ⊆ I+n for n = 0, 1, and(ii) if ηRS

1 ξ , then η ∈ I0 ↔ ξ ∈ I1

Then there is a winning strategy for player ∃ in EFλU (I0, I1).

Trees I0 and I1 satisfying conditions (i) and (ii) above are called S-obedient.The following is Lemma 3.17 in [5].

Lemma 6.12. Suppose S ⊆ Sλκ is stationary in λ. Then there are S-obedient

(λ+, ω + 2)-trees I0 and I1 such that I0 �∼= I1.


Lemma 6.13. Suppose S ⊆ Sλκ is stationary such that player ∃ has a winning strat-

egy in the club game GCκ(Sλκ \S, λ). Then there are (λ+, ω + 2)-trees I0 and I1

such that

I0 �∼= I1 and I0 ≡λU I1

for every U ∈ Uλ.

Proof. Let I0 and I1 be as in Lemma 6.12. By Corollary 5.3, player ∃ has a win-ning strategy in GCκ

U (Sλκ \S, λ) for every U ∈ Uλ. Hence, by Lemma 6.11 and

Lemma 6.12, the claim follows. ��

7. Forcing isomorphisms

In this section we study the problem when a complete countable theory T hasnon-isomorphic models that can be forced to be isomorphic without disturbing theuniverse too much. By this vague expression we intend to describe a common themeof the statements of theorems below. It could mean, for instance, that the forcingextension has no new subsets of a small cardinality, or that stationary subsets of thecardinality of the models are preserved.

This study was motivated by the result due to Baldwin, S. Buechler, Laskowskiand Shelah stating that for classifiable theories it is not possible to have such models[2, p. 1291–2] (problems in this direction are studied further in [10]).

There are two methods how we can find the desired models. First, we can utilizea model A of T of cardinality λ that does not have a universal equivalence tree. Thisapproach is simple and it relies on the results on the existence of such a model. Thedrawback is that these existence results make rather strong assumptions on cardinalarithmetic and that for unsuperstable theories without DOP there need not be sucha model. The following table summarizes the assumptions (Theorems 4.9, 6.2 and6.6 in [9], and Theorem 1.1 in [7]).

Theory AssumptionsUnstable λ<λ = λ > ω

Stable with DOP λ<λ = λ, ∀ξ < λ(ξκ < λ),κ = cf(κ) = λ(T )+ κ<κ(T ) > ω

Superstable with DOP or OTOP λ<λ = λ > ω

The other method makes use of the general non-structure theory developed byShelah. (If T is unsuperstable, then we will also use the method of Section 5.) Thisapproach is technically more involved but the results are better in the sense that wedo not have to make as strong assumptions as in the first approach and it appliesalso to unsuperstable theories without DOP. On the other hand, using this methodwe cannot preserve stationary subsets of λ. The whole idea is that a bistationaryset S is coded into the models and when a club set is forced into the complementof the set S without adding subsets of small cardinality the models will becomeisomorphic. The notion of property M(S) will formalize this idea (Definition 7.14).

By using the latter method we can cover all other cases except the one that mustbe excluded by the result due to Baldwin, Buechler, Laskowski and Shelah, namelysuperstable theory without DOP and OTOP.


7.1. On killing stationary sets

The problem of killing a stationary set is well known and it is studied, e.g., in [3],[1] and [4]. A stationary set S ⊆ κ is killed by adding a club set into its complementS′. There are various constraints that can be put on the partial order P adding aclub set into the stationary set S′. Regarding our intended application the partialorder P must not add new subsets of cardinality < κ , as was indicated above inthe characterization of the property M(S). A stationary set for which there is sucha partial order is necessarily fat [1, p. 644]. The definition of a fat stationary set isas follows: A stationary set S′ ⊆ κ is fat if and only if for every club set C of κ ,S′ ∩ C contains closed sets of arbitrarily large order-types below κ . For instance,the following is proved in [1].

Theorem 7.1 ([1, Theorem 1]). Let κ be either a strongly inaccessible cardinal orthe successor of a regular cardinal µ such that µ<µ = µ. Let S ⊆ κ be a fatstationary set. Then there is a partial order P such that the following hold.

(i) Forcing with P adds a club set C ⊆ S.(ii) Forcing with P does not add new subsets of cardinality < κ .

The theorem makes rather strong assumptions, but on the other hand it applies toany fat stationary set.

Remark 7.2. It is consistent relative to the consistency of the existence of a Mahlocardinal that there is a fat stationary set S ⊆ ω2 into which it is impossible to forcea club set without adding subsets of cardinality < ω2 [4, Section 3]. ��

For our purposes it suffices to have certain kind of fat stationary sets.

Definition 7.3 ([9, Definition 9.3]). Suppose κ is a cardinal. A stationary set S ⊆ κ

is called strongly bistationary if for every α < κ player ∀ does not have a winning

strategy in GCSκ

regα (κ\S, κ).

Lemma 7.4. Suppose κ is a regular cardinal and S ⊆ κ a strongly bistationaryset. Then there is a partial order P such that in every P -generic extension there isa club set C ⊆ κ\S and no subsets of cardinality less than κ are added.

Proof. Let S′ = κ\S and P = {a ⊆ S′ | a is closed} ordered by end-extension.Clearly forcing with P adds a club into S′.

To complete the proof it suffices to show that any function f in a P -genericextension with the domain a cardinal less than κ and the range a subset of V , thefunction f is already in V . So suppose f is a P -name, λ < κ and p ∈ P such that

p � f is a function from λ to V .

We define a strategy σ for player ∀ in GCSκ

regλ (S′, κ) as follows. Along with the

ordinals player ∀ chooses a decreasing sequence 〈pξ | ξ < λ〉 of conditions for hisown purposes. The ordinals βξ , ξ < λ, are the choices of player ∃. Let <∗ be somewell-ordering of P . The choices of player ∀ are as follows:


p0 = p and α0 = sup(p),pξ is the <∗-first condition r which extends pζ for all ζ < ξ such that sup(r) >

sup{βζ | ζ < ξ} and it decides the value of f (ξ),αξ = sup(pξ ).

Since S is strongly bistationary, the strategy σ is not a winning strategy for play-er ∀. Let

⟨〈αξ | ξ < λ〉, 〈βξ | ξ < λ〉⟩ be a play that player ∃ wins against thestrategy σ . Then β = sup{βξ | ξ < λ} ∈ S′. Since sup{sup(pξ ) | ξ < λ} = β,q = (

⋃ξ<λ pξ ) � 〈β〉 is a condition which extends p and decides f . Hence p

forces “f ∈ V ”. ��Remark 7.5. Also the converse of the previous lemma holds. ��Lemma 7.6 ([8]). Suppose κ is a singular cardinal or κ = ω or there is a stationaryset E ⊆ Sκ+

ω with ��κ(E). Then there is a strongly bistationary set S ⊆ Sκ+ω .

Proof. This follows immediately from Theorems 3.4 and 3.8 in [8]. ��Lemma 7.7. Suppose κ is a regular cardinal and S ⊆ Sκ+

κ . If player ∀ does nothave a winning strategy in GCκ(Sκ+

κ \S, κ+), then he does not have a winning

strategy in GCSκ+

regα (κ+\S, κ+) for all α < κ+.

Proof. This follows from [8, Lemma 3.2]. ��Lemma 7.8. Suppose κ is a regular cardinal. Then there is a strongly bistationaryS ⊆ Sκ+

κ .

Proof. This follows from the previous lemma and the fact that player ∀ cannot havewinning strategies both in GCκ(Sκ+

κ \S, κ+) and GCκ(S, κ+). ��Definition 7.9. Let κ be a regular cardinal. We define F [κ+] to be the collection ofsubsets S ⊆ κ+ such that there is a winning strategy for player ∃ in GCκ(S, κ+).

Lemma 7.10. Suppose κ is a regular cardinal. Then F [κ+] is a non-trivial, non-principal, and κ+-complete filter.

Proof. Suppose S ∈ F [κ+]. Since player ∃ has a winning strategy in GCκ(S, κ+),S is stationary. Hence F [κ+] cannot be trivial or principal. Clearly if S ⊆ S′ ⊆ κ+,then S′ ∈ F [κ+]. So it suffices to show that F [κ+] is κ+-complete. Let Si ∈ F [κ+]and let wi be a winning strategy for player ∃ in GCκ(Si, κ

+) for i < κ . Choose afunction f : κ → κ with |f−1{i}| = κ for each i < κ . Define a strategy w forplayer ∃ by

w(α0, . . . , αi) = wf (i)(α)

where α = 〈αj | j ≤ i ∧ f (j) = f (i)〉.Let 〈α, β〉 be a play in which player ∃ plays according to the strategy w. For

i < κ , let Ai = {αij | j < κ} = f−1{i}. Now,

〈〈αij | j < κ〉, 〈wi(α

i0, . . . , αi

j ) | j < κ〉〉is a play won by player ∃. Hence sup {αi

j | j < κ} = sup {αi | i < κ} = α ∈ Si .Therefore α ∈ S =⋂

i<κ Si , and S ∈ F [κ+]. ��


Since κ+ cannot be a measurable cardinal, F [κ+] is not an ultra-filter over κ+,and hence we have the following corollary.

Corollary 7.11. Suppose κ is a regular cardinal. Then there is a set S ⊆ κ+ suchthat there is no winning strategy for player ∃ in either GCκ(S, κ+) orGCκ(κ+\S, κ+).

7.2. Using models with no universal equivalence trees

Suppose that a theory T has a model A of cardinality κ for which no κ-tree ofcardinality at most κ is an universal equivalence tree. Let U be a κ-tree of heightκ , and let B be a model of T such that A ≡2

U B and A �∼= B. Then we can makethe models isomorphic by forcing a κ-branch into the tree U . We just have to pickthe right tree so that we do not add subsets of cardinality < κ . This can always bedone, and under some additional assumptions we can preserve stationary subsets ofκ and all cardinals as well. The following definition and lemma show the generalframework, and the table above (on page 106) shows when a theory has a modelwith no universal equivalence tree.

Definition 7.12 ([9, Definition 2.7]). Let A be a τ -structure of cardinality κ . A treeU is a universal equivalence tree for A if and only if U is a κ-tree and for everyτ -structure B of the cardinality κ the following holds:

if A ≡2U B, then A ∼= B.

Lemma 7.13. Suppose that T is complete theory, A is a model of T of regularcardinality κ and no (κ+, κ)-tree is a universal equivalence tree for A. If

(i) κ is a successor cardinal, or(ii) there is a κ-Suslin tree,

then there are a partial order P and a model B of T of cardinality κ such that

(iii) A �∼= B,(iv) in every P -generic extension A and B are isomorphic,(v) forcing with P does not add new subsets of cardinality less than κ ,

(vi) if (ii) holds, then P can be chosen so that it preserves stationary subsetsof κ .

Proof. Choose the partial order P and a κ-tree U as follows:

Suppose that (i) holds. Let κ = µ+. If µ is regular, then, by Lemma 7.8, thereis a strongly bistationary set S ⊆ Sκ

µ. If µ is singular, then, by Lemma 7.6,there is a strongly bistationary S ⊆ Sκ

ω. Let S′ = κ\S. Let P be a partial orderthat forces a club set into S′ without adding subsets of cardinality < κ . ByLemma 7.4, such a partial order exists. Let U be the tree of closed subsets ofS′ ordered by end-extension. Clearly, U is a (κ+, κ)-tree.If (ii) holds, let U be a κ-Suslin tree and let P = U .


Since U is not a universal equivalence tree for A, it is possible to pick a model Bof cardinality κ such that A ≡2

U B and A �∼= B (we may assume that the universesare disjoint). Let σ be a winning strategy for player ∃ in EF2

U(A, B).Let G be a P -generic set. Since P is κ-distributive, σ is still a strategy for

player ∃ in V [G]. Let 〈ti | i < κ〉 be a κ-branch in U and {xi | i < κ} anenumeration of A ∪B. Every initial segment of the play

⟨〈〈xi, ti〉 | i < κ〉, 〈σ(〈x0, t0〉, . . . , 〈xi, ti〉) | i < κ〉⟩

is in V . Hence the mapping determined by the play is the required isomorphismfrom A onto B. ��

7.3. Using Ehrenfeucht-Mostowski models

In this subsection we show how we can, by using Ehrenfeucht-Mostowski models,build non-isomorphic models that can be forced to be isomorphic. The structure ofthe proofs of the different cases are always the same. First we define two distinctindex models. A stationary set S is used in the construction of the index models.Linear orders can be used as index models for other cases except for unsuperstabletheories. Because the stationary set is coded – in some sense – into the index mod-els, the corresponding Ehrenfeucht-Mostowski models will be non-isomorphic. Onthe other hand, there is a set of partial isomorphisms having extension propertieswhich ensure that if a club set is forced into the complement of the stationary setS, the models will be isomorphic provided that no subsets of small cardinality areadded. The following definition and lemma give a framework for this.

Definition 7.14. Assume λ is an uncountable regular cardinal and S ⊆ λ is a sta-tionary set. We say that a pair of models A, B has the property M(S) if there aremodels Aα and Bα , α < λ, a set F of isomorphisms and a function G : F → λ\Ssuch that the following hold.

(i) For α < β < λ, Aα ⊆ Aβ and Bα ⊆ Bβ , and if α is a limit ordinal, thenAα =

⋃ξ<α Aξ and Bα =

⋃ξ<α Bξ .

(ii) The models A =⋃α<λ Aα and B =⋃

α<λ Bα are of cardinality λ.(iii) The set F is non-empty and for all f ∈ F , f : AG(f ) → BG(f ).(iv) For all f ∈ F and α < λ, there is g ∈ F with f ⊆ g and G(g) ≥ α.(v) If 〈fi | i < α〉 is an increasing sequence of elements of F with

⋃i<α G(fi) ∈

λ\S, then⋃

i<α fi ∈ F .

Lemma 7.15. Let T be a theory and λ an uncountable regular cardinal. Supposethat S ⊆ λ is stationary in λ and that a partial order PS forces a club set into thecomplement of S without adding subsets of cardinality < λ. Suppose furthermorethat A and B are models of T of cardinality λ and that they have the propertyM(S). Then in every PS-generic extension the models A and B are isomorphic.

Proof. Let Aα and Bα , α < λ, F and G be as in Definition 7.14. Let H be a PS-generic set. Work in V [H ]. Let C = {ci | i < λ} ⊆ λ\S be a club set enumerated


in increasing order. Choose increasing continuous sequences 〈fi | i < λ〉 and〈βi | i < λ〉 such that

fi ∈ F and βi ∈ C,

βi ≥ G(fi),

G(fi+1) > βi.

This is possible since every initial segment of the sequences is in V and C ⊆ λ\Sis a club set. The function

f =⋃

i<λ

fi

is the required isomorphism from A onto B. ��Continuously we make use of the following simple lemma

Lemma 7.16. Suppose that models I0 and I1 are of cardinality λ and a template �

is proper for I0 and I1. Let A and B be the corresponding Ehrenfeucht-Mostowskimodels with skeletons 〈as | s ∈ I0〉 and 〈bs | s ∈ I1〉. Suppose further that themodels I0, I1 have the property M(S). Then the models A, B have the propertyM(S).

Proof. Let sequences 〈I iα | α < λ〉, i = 0, 1, the set F of isomorphisms and the

function G : F → λ\S witness that the models I0, I1 have the property M(S). Forα < λ, let

Aα = hull(A, {as | s ∈ I 0

α }),

Bα = hull(B, {bs | s ∈ I 1

α }).

Let F ′ be the set of functions

{f ′ | f ∈ F }where f ′ is as in 3.5(i). By 3.5(i), the functions in F ′ are isomorphisms. Define G′by G′(f ′) = G(f ). Clearly the sequences 〈Aα | α < λ〉, 〈Bα | α < λ〉, the set F ′and the function G′ satisfy Definition 7.14. (7.16)��

7.3.1. Construction of linear orders Below we construct a linear order which willbe the basic ingredient in the construction of more complex linear orders, whichwe will use as index models in the sequel. The history of this kind of constructionsdates back to [12].

Lemma 7.17. Suppose κ is a cardinal. Then there is a linear order ηκ of cardinalityat most κ+ such that cf∗(ηκ) = ω and

ηκ∼= ηκ + ηκ, (6)

ηκ∼= ηκ · α + ηκ, α ≤ κ, (7)

ηκ∼= ηκ · κ + ηκ · (ω1)

∗. (8)


Proof. We construct in L a linear order η that has the desired properties in L. Theproperties stated in the lemma are upward absolute for transitive models of ZFC,therefore η satisfies the properties also in V .

Work in L. Let µ = (ω1)V . First we define a linear order θ . Let the universe of

θ be

{f ∈ <ω

(Q ∪ (κ + µ∗)

) | ∃n < ω(n > 0 ∧ dom(f ) = 2n ∧

∀i < n(f (2i) ∈ Q ∧ f (2i + 1) ∈ κ + µ∗))}

.

We order θ lexicographically.

Claim 7.18. The coinitiality of θ is ω and it has the following properties:

θ ∼= (1+ θ) · (κ + µ∗) ·Q, (9)

θ ∼= θ + θ, (10)

θ ∼= θ + 1+ θ, (11)

θ ∼= θ + (1+ θ) · κ + (1+ θ) · µ∗. (12)

Proof. Define the mapping g : θ → (1+ θ)× (κ + µ∗)×Q by

g(f ) = 〈f ′, f (1), f (0)〉

where f ′ is such that dom(f ′) = dom(f ) − 2 and f ′(i) = f (i + 2) for everyi ∈ dom(f ′). Clearly g is an isomorphism from θ onto (1+ θ) · (κ +µ∗) ·Q. ThusEquation (9) holds.

It follows from Q ∼= Q+Q and (9) that θ ∼= θ + θ .Using the Equations (9) and (10) and the facts Q ∼= Q+ 1+Q, κ ∼= 1+ κ , we

get

θ ∼= (1+ θ) · (κ + µ∗) · (Q+ 1+Q)

∼= θ + (1+ θ) · (κ + µ∗)+ θ

∼= θ + 1+ θ + (1+ θ) · (κ + µ∗)+ θ

∼= θ + 1+ θ.

By the observations above and the fact that µ∗ = µ∗ + 1, we have

θ ∼= (1+ θ) · (κ + µ∗) · (Q+ 1+Q)

∼= θ + (1+ θ) · (κ + µ∗)+ θ

∼= θ + (1+ θ) · (κ + µ∗)+ 1+ θ + θ

∼= θ + (1+ θ) · κ + (1+ θ) · µ∗ + 1+ θ

∼= θ + (1+ θ) · κ + (1+ θ) · µ∗.

For each i < ω, let fi ∈ θ be such that fi(0) = −i. Clearly the sequence〈fi | i < ω〉 is coinitial in θ . (7.18)��


Now we are ready to define η. Its universe is

{f ∈ ω(θ ∪ κ) | ∀i < ω(f (2i) ∈ θ ∧ f (2i + 1) ∈ κ)},and we order it lexicographically.

By similar equations as in the proof of the claim above we can show that equa-tions (6)–(8) hold and cf∗(ηκ) = ω.

Since η is defined in L, it is of cardinality (κω)L which is certainly at most κ+.(7.17)��

Note that if κ is of uncountable cofinality then the linear order η is ofcardinality κ .

The �-models which we define below were inspired by those that J. Conwayintroduced to classify all ω1-like dense linear orders [12, p. 58].

Definition 7.19. Suppose that λ is an ordinal and η a linear order. For a subset A

of λ we define the linear order �λ(A, η) by

�λ(A, η) =∑

α<λ

τ(α, A, η)

where

τ(α, A, η) ={

η if α /∈ A

η · (ω1)∗ otherwise.

For α < β < λ we define the linear order �λ(A, η, α, β) by

�λ(A, η, α, β) =∑

α≤γ<β

τ(γ, A, η).

Let κ be a cardinal. By �λ,κ(A) we denote the linear order �λ(A, ηκ). Whenκ is clear from the context we write briefly �λ(A) for �λ,κ(A). Similarly, by�λ,κ(A, α, β) we denote the linear oder �λ(A, ηκ, α, β), and if κ is clear from thecontext we write �λ(A, α, β) for �λ,κ(A, α, β).

Lemma 7.20. Suppose κ is a cardinal and η = ηκ as in Lemma 7.17, S ⊆ Sκ+κ

and α ≤ β < κ+. Then �κ+,κ (S, α, β + 1) ∼= η if α /∈ S.

Proof. The proof is by induction on β.β = α: Then, by Definition 7.19, �κ+(S, α, α + 1) ∼= η.β is a successor ordinal: Then β /∈ S as S contains only limit ordinals. Thus

�κ+(S, α, β + 1) ∼= �κ+(S, α, β)+�κ+(S, β, β + 1)

∼= η + η

∼= η.

β ∈ S is a limit ordinal: Since S ⊆ Sκ+κ , we can choose an increasing and

continuous sequence 〈cξ | ξ < κ〉 cofinal in β such that c0 > α and each cξ /∈ S.


By induction hypothesis we have�κ+(S, α, c0+1) ∼= η, �κ+(S, cξ+1, cξ+1+1) ∼=η and �κ+(S, cξ , cξ+1 + 1) ∼= η as cξ , cξ + 1 /∈ S. Hence

�κ+(S, α, β + 1) ∼= η · κ + η · (ω1)∗ ∼= η.

The latter isomorphism holds by (8).β /∈ S is a limit ordinal: This goes exactly as the previous case but now we get

�κ+(S, α, β + 1) ∼= η · cf(β)+ η ∼= η.

This time the latter isomorphism follows from (7). ��Lemma 7.21. Suppose κ is a cardinal and η = ηκ as in Lemma 7.17, S1, S2 ⊆ Sκ+

κ ,and S = κ+\(S1 ∪ S2) is κ-stationary. Then the pair �κ+κ(S1), �z+,z(S2) has theproperty M(S1 ∪ S2).

Proof. As S is κ-stationary and Sκ+<κ ⊆ S, there is a continuous sequence sα =

{cαi | i < α} for every α < κ+ such that each cα

i ∈ S, cα0 = 0 and cα

i is a successorordinal if i is a successor ordinal. By Lemma 7.20,

�κ+(S1, ci, ci+1) ∼= η ∼= �κ+(S2, ci, ci+1)

for every i < α. Hence there is an isomorphism fsα from �κ+(S1, 0, sup(sα))

onto �κ+(S2, 0, sup(sα)). Define Aα = �κ+(S1, 0, α) and Bα = �κ+(S2, 0, α),α < κ+. Let F be the set

{fsα | sα ⊆ S continuous of order type α and sup(sα) ∈ S}.of isomorphisms. Define the function G : F → S by fsα �→ sup(sα). Clearly themodels Aα and Bα , α < κ+, the set F , and the function G witness that �κ+(S1)

and �κ+(S2) have the property M(S1 ∪ S2). ��

7.3.2. Theories which are unstable or superstable

Theorem 7.22. Let T be a complete theory in a countable vocabulary τ and let κ

be a regular cardinal. Suppose that one of the following conditions hold.

(i) The theory T is unstable.(ii) The theory T is superstable with DOP and κ+ ≥ 2ω.

(iii) The theory T is superstable with OTOP.

Then there are non-isomorphic models A, B |= T of cardinality κ+ and a partialorder P that forces the models A and B to be isomorphic without adding subsetsof cardinality κ .

Proof. We only prove the result for unstable theories, i.e., the case (i). The other twocases can be proved in a similar fashion, only the linear orders, which are denotedbelow by Aα , must be adjusted for the needs of [14, Fact 2.5B] and [14, Fact XII4.16].

Suppose S ⊆ Sκ+κ is stationary such that there is no winning strategy for player ∃

in GCκ(S, κ+). Let η be the linear order given by Lemma 7.17. Choose stationary


sets Sα ⊂ S, α < κ++, such that for every α < β < κ++, the set Sα!Sβ isstationary. For each α < κ++, let Iα = Q ·�κ+(Sα) and define Aα = Iα

∗ · κ+.By [15, Lemma 1.2], there is Aα = EM(Aα) |= T of cardinality κ+ for each

α < κ++. Using [13, Lemma III 3.13] and [13, Lemma III 3.15] and the fact thatfor every α < β < κ++, the set Sα!Sβ is stationary in κ+, it can be showed thatfor some α < β < κ++, the models Aα and Aβ are non-isomorphic.

By Lemma 7.21, the linear orders �κ+(Sα), �κ+(Sβ) have the property M(Sα∪Sβ). Hence the linear orders Aα and Aβ have the property, too. By Lemma 7.16,the models Aα , Aβ have the property M(Sα ∪ Sβ).

Since there is no winning strategy for player ∃ in GCκ(S, κ+), there is a partialorder P that forces a κ-club into Sκ+

κ \S without adding subsets of cardinality atmost κ (see [4, Lemma 3.1]). Let G be a P -generic set. Then, by Lemma 7.15,Aα∼= Aβ in V [G]. ��

7.3.3. Unsuperstable theories This case is trickier than the previous cases. Wecannot just repeat the proof of Theorem 7.22 because in that proof the index modelsmust be linear orders and by Theorem 3.6 the index models are ordered trees ofsmall height for unsuperstable theories.

For a regular cardinal κ > ω1 we can build the required non-isomorphic modelsof cardinality κ+ by using the construction of

Section 5 if κω = κ . For a singular cardinal κ we can use the construction in[9, Section 8], which is essentially the construction presented in [17], to build therequired non-isomorphic models of cardinality κ+. In this case we need a stronglybistationary set S ⊆ Sκ+

ω . In order to be able to prove that the constructed modelshave the property M(S) (Lemma 7.29) we give the construction below.

Definition 7.23 ([9, Definition 3.2]). Let κ be an ordinal. By θκ we denote the linearorder with universe <ωκ ordered lexicographically.

Lemma 7.24 ([9, Lemma 8.17]). Suppose κ is a cardinal and θκ as in Defini-tion 7.23. Then there is E ⊆ θκ of cardinality κ such that for any a, b ∈ E

there is an automorphism ga,b of θκ which maps a to b.

Definition 7.25 ([9, Definition 8.4]). Let A be an arbitrary set of cardinality atmost κ . A sequence A = 〈Aα | α < κ〉 is a κ-representation of A if Aα ⊆ A,|Aα| < κ , Aα ⊆ Aβ for all α ≤ β < κ , Aα =

⋃β<α Aβ for all limit α < κ , and

A =⋃α<κ Aα .

Definition 7.26 ([9, Definition 8.4]). Suppose κ is a cardinal, S ⊆ Sκω and S =

〈ηα | α ∈ S〉 where each ηα is an increasing cofinal sequence in α of order type ω.The ordered tree I (κ, S) is defined as follows. The universe of I (κ, S) is

<ωκ ∪ {ηα | ηα ∈ S}.The relations are defined as in Definition 3.2.

Definition 7.27 ([9, pp. 238–239]). Suppose κ is a cardinal and the ordered treesIi ∈ Kω

tr (κ), i < 2, are such that |Ii | = κ and <ωκ ⊆ Ii . Let Ii = 〈I iα | α < κ〉 be


a κ-representation of Ii . Let R(I0, I1) = {I iα | i < 2 ∧ α < κ}. Let E ⊆ θκ be as

in Lemma 7.24, c ∈ E and g : R(I0, I1)→ E\{c} a bijection. Let <ωκ be orderedso that the order type is κ , and let θ = <ωκ · θκ . The ordered tree J (c, g, I0, I1)

is defined as follows. The universe of J (c, g, I0, I1) consists of functions η ∈ ≤ωθ

such that one of the following holds:

(i) η ∈ <ωθ .(ii) There is s ∈ I0 such that dom(s) = ω and for all n < ω, η(n) = 〈s�(n+1), c〉.

(iii) There are m < ω, R ∈ R(I0, I1) and s ∈ R with dom(s) = ω such that forall finite n ≥ m, η(n) = 〈s�(n+ 1), g(R)〉.

The relations of J (c, g, I0, I1) are defined as in Definition 3.2.

The following definition and lemma, which show the existence of the requiredset of partial isomorphisms, have their roots in [9], Definition 8.19 and Lemma8.20, respectively.

Definition 7.28. (i) Let I ∈ Kωtr and A ⊆ I . Define

b(A) = {η ∈ I | ∃ρ ∈ I (η ∈ succ(ρ) ∧ succ(ρ) ∩ A �= ∅)} ∪ A,

i.e., the closure of A under brotherhood.(ii) Let κ , Ii , Ii , i < 2, c and g be as in Definition 7.27. Let J0 = J (c, g, I0, I1)

and J1 = J (c, g, I1, I0). We define the set F(J0, J1) by f ∈ F(J0, J1) if andonly if the following conditions hold:(a) f ∈ Part(J0, J1),(b) there exists α < κ with dom(f ) = b(J 0

α ) and ran(f ) = b(J 1α )

where

J iα = Ji ∩

{η ∈ ≤ωθ | ∀n ∈ dom(η)

(ran(1st(η(n))) ⊆ α

)}

for i < 2. The ordinal α < κ with dom(f ) = b(J 0α ) is denoted by α(f ).

Lemma 7.29. Suppose κ is an uncountable regular cardinal, S ⊆ Sκω, and S is as

in Definition 7.26. Let I0 = I (κ, S) and I1 = I (κ, 〈〉) = <ωκ . Let Ii = 〈Ii ∩≤ωα |α < κ〉, i < 2, and let c and g be as in Definition 7.27, and J0 = J (c, g, I0, I1)

and J1 = J (c, g, I1, I0).

(i) For every f ∈ F(J0, J1) and β < κ greater than α(f ) there is f ′ ∈ F(J0, J1)

with f ⊆ f ′ and α(f ′) = β.(ii) Assume {fξ | ξ < γ } ⊆ F(J0, J1) is an increasing chain with sup{α(fξ ) |

ξ < γ } ∈ κ\S. Then⋃

ξ<γ fξ ∈ F(J0, J1).

Proof. (i) Let

C = {ν ∈ J 0

β | ∀n ∈ dom(ν)(2nd(ν(n)) = c

)},

A′′ = b(C)\dom(f ),

A′ = b(J 0β )\A′′,


and d = g(I 0β ). The elements in C are called c-nodes. First we define a function f ′′

such that f ′′ ⊇ f and dom(f ′′) = dom(f ) ∪ A′′. Let ν ∈ A′′ and let η be amaximal node in dom(f ) which is below ν. Let sn and bn be such that

ν = η � 〈〈sn, bn〉 | m ≤ n < α〉

where α = dom(ν) and m = dom(η). We define

f ′′(ν) = f (η) � 〈〈sn, gc,d(bn)〉 | m ≤ n < α〉.

Then we extend f ′′ to a function f ′ such that dom(f ′) = b(J 0β ) as follows. Let

ν ∈ A′ and let η be a maximal node in dom(f ′′) which is below ν. Let sn and bn

be such that

ν = η � 〈〈sn, bn〉 | m ≤ n < α〉

where α = dom(ν) and m = dom(η). We define

f ′(ν) = f ′′(η) � 〈〈sn, bn〉 | m ≤ n < α〉.

Clearly dom(f ′) = b(J 0β ).

Since the function f ′ does not change ordinals ≥ α in nodes (i.e., for allν ∈ dom(f ′) and i ∈ dom(ν), if ν�i /∈ dom(f ) then 1st(ν(i)) = 1st

(f ′(ν)(i)

)),

we have ran(f ′) ⊆ b(J 1β ).

To see b(J 1β ) ⊆ ran(f ′), suppose that

ν = 〈〈sn, bn〉 | n < δ〉 ∈ b(J 1β )

where δ ≤ ω. We may assume that every proper initial segment of ν is in ran(f ′).Suppose first that δ < ω. Clearly δ > 0. Let η ∈ b(J 0

β ) be such that f ′(η) =ν�(δ − 1). If η is a c-node, then e = g−1

c,d (bδ−1) satisfies η � 〈〈sδ−1, e〉〉 ∈ A′′.By the definition of the function f ′′, we have ν = f ′′(η � 〈sδ−1, e〉). Thus ν ∈ran(f ′′) ⊆ ran(f ′). If η is not a c-node, then ν = f ′(η � 〈sδ−1, bδ−1〉). Againν ∈ ran(f ′).

Suppose then δ = ω. By 7.27(iii) and the fact that I1 = <ωθ , there are γ ≤ β,s ∈ I 0

γ and m < ω such that sn = s�(n+1) and bn = g(I 0γ ) for every finite n > m.

Since every proper initial segment of ν is in ran(f ′), there are

ηn = 〈〈si, t (i)〉 | i ≤ n〉 ∈ dom(f ′)

for n < ω with f ′(ηn) = ν�(n+1). Let η =⋃n<ω ηn. It follows from s ∈ I 0

γ ⊆ I 0β

that η ∈ J 0β . Let k < ω be maximal with ηk ∈ dom(f ) and ηk+1 /∈ dom(f ). Let

l ≤ ω be maximal such that for every i < l, t (i) = c. If l = ω, then

f ′(η) = f (ηk) � 〈〈sn, d〉 | k < n < ω〉 =⋃

i<ω

ν�i = ν.


If l < ω, then

f ′(η) = f (ηk) � 〈〈sn, gc,d(t (n))〉 | k < n ≤ l〉 �〈〈sn, t (n)〉 | l < n < ω〉 =

⋃

i<ω

ν�i = ν.

Thus ν ∈ ran(f ′).Clearly, f ′ is a partial isomorphism from b(J0) to b(J1). The only concern here

is the < relation. But because dom(f ) and dom(f ′) are closed under brotherhood,the isomorphism condition is satisfied with respect to the < relation.

(ii) Let f = ⋃ξ<γ fξ and α = sup{α(fξ ) | ξ < γ }. Obviously f ∈

Part(J0, J1), and dom(f ) ⊆ b(J 0α ) and ran(f ) ⊆ b(J 1

α ). Suppose η ∈ b(J 0α ).

If dom(η) < ω, then there is ξ < γ with η ∈ dom(fξ ). If dom(η) = ω,

sup{ran

(1st(η(n))

) | n < ω} = β < α.

Hence there is ξ < γ withη ∈ b(J 0α(fξ ))which yieldsη ∈ dom(f ). Thus dom(f ) =

b(J 0α ). Similarly we can show ran(f ) = b(J 1

α ). (7.29)��Lemma 7.30 ([9]). Let κ be an uncountable regular cardinal, and τ , τ1, T , T1 and� be as in Theorem 3.6, and |τ1| < κ . Suppose S ⊆ Sκ

ω is stationary. Let J0 andJ1 be as in Lemma 7.29. Then EM(J0, �) �∼= EM(J1, �).

Proof. This follows directly from Theorem 8.13, Lemma 8.14 and Lemma 8.20 in[9]. ��Lemma 7.31. Suppose that the assumptions of Lemma 7.29 hold. Then the orderedtrees J0 and J1 have the property M(S).

Proof. By the previous lemma, J0 �∼= J1. For i = 0, 1, define continuous sequences〈J i

α | α < κ〉 such that J i0 = ∅ and for every α < κ it holds that

J iα+1 = Ei

α+1 ∩ Ji

where

Eiα =

{η ∈ ≤ωθ | ∃n ∈ dom(η)

(ran(1st(η(n))) ⊆ α

)}.

It follows that for each α ∈ Sκlim\S, we have

J iα =

⋃

β<α

(Eiβ ∩ Ji).

Let F be the set of isomorphisms

{f ∈ F(J0, J1) | α(f ) ∈ κ\S},and define the function G : F → κ\S by

G(f ) = α(f ).

By Lemma 7.29, these objects witness that J0, J1 have the property M(S). ��


Theorem 7.32. Let T be a countable complete unsuperstable theory and κ acardinal. Suppose that one of the following hold:

(i) The cardinal κ is singular or κ = ω or there is a stationary set S ⊆ Sκ+ω with

��κ(S).(ii) The cardinal κ is regular and κω = κ .

Then there are models A, B |= T of cardinality κ+ such that A �∼= B and a par-tial order P that does not add subsets of cardinality at most κ such that in everyP -generic extension A ∼= B.

Proof. (i) By Lemma 7.6, there is a strongly bistationary S ⊆ Sκ+ω . Let P be the

partial order given by Lemma 7.4.LetJ0 andJ1 be as in Lemma 7.29. SinceS is stationary, we have, by Lemma 7.30,

EM(J0, �) �∼= EM(J1, �). By Lemma 7.31 and Lemma 7.16, the models EM(J0, �)

and EM(J1, �)have the propertyM(S). By Lemma 7.15, EM(J0, �) ∼= EM(J1, �)

in every P -generic extension.(ii) By Corollary 7.11, choose a κ-bistationary set S′ ⊆ Sκ+

κ such that player ∃does not have a winning strategy either in GCκ(S′, κ+) or GCκ(Sκ+

κ \S′, κ+). ByLemma 4.4, there is an ω-club-guessing sequence for S′ or Sκ+

κ \S′. Let S be eitherof them for which there is an ω-club-guessing sequence. By [4], let P be a partialorder which forces a club set into the complement of S without adding new subsetsof cardinality ≤ κ .

Let J0 = J S0 and J1 = J S

1 be as in Definition 5.13, and A = EM(J0, �) andB = EM(J1, �). By Lemma 5.14, A �∼= B.

Let 〈aρ | ρ ∈ J0〉 and 〈bρ | ρ ∈ J1〉 be the skeletons of A and B, respectively.Define continuous sequences 〈J i

α | α < κ+〉 such that for every α < κ+ andi = 0, 1,

J iα+1 = Ji(α + 1)

where Ji(α + 1) is as in 5.6(iv). (Note that the sequence 〈Ji(α) | α < κ+〉 is notcontinuous.) For α < κ+, let

Aα = hull(A, {aρ | ρ ∈ J 0α }),

Bα = hull(B, {bρ | ρ ∈ J 1α }).

Let F be the set of functions

{f ∈ GJ0,J1 | ∃α ∈ κ+\S(dom(f ) = J 0α )}.

By Lemma 5.11, the set F satisfy Conditions (iii)-(v) of Definition 7.14 for J0, J1.Hence J0, J1 have the property M(S). Therefore, by Lemma 7.16, the models A,B have the property M(S). (7.32)��Conclusion 7.33. Suppose T is a countable complete theory which is not classifi-able. Then there are arbitrary large cardinals λ such that there are two non-iso-morphic models A, B |= T of cardinality λ which can be forced to be isomorphicwithout adding subsets of cardinality < λ.


Proof. Follows immediately from theorems 7.22 and 7.32. ��Note that by the proof of Theorem 7.22 we can build 2λ pairwise non-isomor-

phic but potentially isomorphic models of cardinality λ where λ is the successor ofa regular uncountable cardinal κ . With some additional work we can have a similarresult for unsuperstable theories if λ is the successor of a singular cardinal.

8. References

1. Abraham, U., Shelah, S.: Forcing closed unbounded sets. J. Symbolic Logic 48, 643–657(1983)

2. Baldwin, J.T., Laskowski, M.C., Shelah, S.: Forcing isomorphism. J. Symbolic Logic58, 1291–1301 (1993)

3. Baumgartner, J.E., Harrington, L.A., Kleinberg, E.M.: Adding a closed unbounded set.J. Symbolic Logic 41, 481–482 (1976)

4. Huuskonen, T., Hyttinen, T., Rautila, M.: On the κ-cub game on λ and I [λ]. Arch. Math.Logic 38, 549–557 (1999)

5. Hyttinen, T., Shelah, S.: Constructing strongly equivalent nonisomorphic models forunsuperstable theories. Part A. J. Symbolic Logic 59, 984–996 (1994)

6. Hyttinen, T., Shelah, S.: Constructing strongly equivalent nonisomorphic models forunsuperstable theories. Part B. J. Symbolic Logic 60, 1260–1272 (1995)

7. Hyttinen, T., Shelah, S.: Constructing strongly equivalent nonisomorphic models forunsuperstable theories. Part C. J. Symbolic Logic 64, 634–642 (1999)

8. Hyttinen, T., Shelah, S., Tuuri, H.: Remarks on strong nonstructure theorems. NotreDame J. Formal Logic 34, 157–168 (1993)

9. Hyttinen, T., Tuuri, H.: Constructing strongly equivalent nonisomorphic models forunstable theories. Ann. Pure Appl. Logic 52, 203–248 (1991)

10. Laskowski, M.C., Shelah, S.: Forcing isomorphism. II. J. Symbolic Logic 61, 1305–1320 (1996)

11. Mitchell, W.: Aronszajn trees and the independence of the transfer property. Ann. Math.Logic 5, 21–46 (1972)

12. Nadel, M., Stavi, J.: L∞λ-equivalence, isomorphism and potential isomorphism. Trans.Amer. Math. Soc. 236, 51–74 (1978)

13. Shelah, S.: Non structure theory. In preparation14. Shelah, S.: Classification theory and the number of nonisomorphic models. North-Hol-

land Publishing Co., Amsterdam, second edition, 199015. Shelah, S.: The number of non-isomorphic models of an unstable first-order theory.

Israel J. Math. 9, 473–487 (1971)16. Shelah, S.: Classification of nonelementary classes. II. Abstract elementary classes.

In J. T. Baldwin, editor, Proceedings of the USA–Israel Conference on ClassificationTheory, volume 1292, Lecture Notes in Mathematics. Springer-Verlag, 1987

17. Shelah, S.: Existence of many L∞,λ-equivalent, nonisomorphic models of T of powerλ. Ann. Pure Appl. Logic 34, 291–310 (1987). Stability in model theory (Trento, 1984)

18. Shelah, S.: There are Jonsson algebras in many inaccessible cardinals. In: CardinalArithmetic, volume 29, Oxford Logic Guides. Oxford University Press, 1994

On potential isomorphism and non-structure

Documents

Transcript of On potential isomorphism and non-structure