However, neither of theseobservations improve the statement of 1.14.1.14 Theorem. If T is a complete theory in a vocabulary L with |L| ≤2ωand T has either OTOP or DOP then there are models M1 and M2 of Twith cardinality the continuum that are not isomorphic but are potentiallyisomorphic.Proof.

By Theorem 1.8 we may assume that T is superstable. By Propo-sition 1.11 and Remark 1.13(i) there is a model M of a theory T1 ⊇T ina Skolemized language L1 ⊇L containing a set of L1-order indiscernibles10

{aη : η ∈Q≤ω} and an ˆLλ+,ω-formula Φ(x, y) so that Φ(aη, aν) holds inM if and only if η is lexicographically less than ν.Further, the state-ments “M |= Φ(aη, aν)” and “M |= ¬Φ(aη, aν)” are preserved under anyccc forcing.Note that this L1-order indiscernibility certainly implies L1-tree-indiscernibility in the sense of Definition 1.7.Thus, the construction of potentially isomorphic but not isomorphic mod-els proceeds as in the last few paragraphs of the proof of Theorem 1.8 oncewe establish the following claim.Claim. For any ν ∈Qω there is a collection pν(x) of boolean combina-tions of Φ(x, a) as a ranges over Y such that for any η ∈Qω and any L1-termf, if f(aη1, .

. .

aηn) realizes pν in M then some ηi = ν.Proof. The conjunction of the Φ(x; aν|n⌢<ν(n)+1>) and ¬Φ(x; aν|n⌢<ν(n)+1>)that define the ‘cut’ of aν will constitute pν.

Now if ν is not among the ηichoose any n such that η1|n, η2|n, . .

. ηk|n, ν|n are distinct.Then the se-quences ⟨η1, .

. .

ηk, ν|n⌢⟨ν(n) + 1⟩⟩and ⟨η1, . .

. ηk, ν|n⌢⟨ν(n) −1⟩⟩have thesame type in the lexicographic order soM |= Φ(f(aη1, .

. .

aηk); aν|n⌢<ν(n)+1>) ↔Φ(f(aη1, . .

. aηk); aν|n⌢<ν(n)−1>).Thus, f(aη1, .

. .

aηk) cannot realize pν.1.15 Remarks.i) Note that in Theorem 1.8 we were able to use anyexpansion of T as T1 so the result is actually for PC∆-classes.InTheorem 1.14 our choice of T1 was constrained, so the result is truefor only elementary as opposed to pseudoelementary classes. The caseof unstable elementary classes could be handled by the second methodthus simplifying the combinatorics at the cost of weakening the result.ii) While we have dealt only with models and theories of cardinality 2ω,the result extends immediately to models of any larger cardinality andstraightforwardly to theories of cardinality κ with κω = κ.2Classifiable examplesWe begin by giving an example of a classifiable theory having a pair of non-isomorphic, potentially isomorphic models.

We then extend this result to aclass of weakly minimal theories.11

Let the language L0 consist of a countable family Ei of binary relationsymbols and let the language L1 contain an additional uncountable set ofunary predicates Pη. We first construct an L0-structure that is rigid butcan be forced by a ccc-forcing to be nonrigid.

Our example will be in thelanguage L0 but we will use expansions of the L0-structures to L1-structuresin the argument.We now revise the definitions leading up to the notion of an amenablestructure in Section 1 by replacing the underlying structure on Q≤ω by onewith universe 2ω. In particular, Dη, Sη, and C are now being redefined.2.1 Notation.i) For η ∈2ω, let Dη = {σ ∈2ω : σ(2n) = η(n)} and Sη = {σ ∈Dη : σ(2n + 1) is 0 for all but finitely many n} ∪{bη}, where bη is anyelement of Dη satisfying bη(2n + 1) = 1 for infinitely many n.LetC = ∪η∈2ωSη.ii) Let M∗be the L1-structure with universe 2ω where Ei(σ, τ) holds ifσ|i = τ|i, and the unary relation symbol Pη holds of the set Sη.

LetM1 be the L1-substructure of M∗with universe C.iii) Any subset A of C inherits a natural L1 structure from M1 with Pηinterpreted as Sη ∩A.2.2 Definition. An L1-substructure M0 of M1 is amenable if for all η ∈2ω,all n ∈ω and all s ∈2n, if there is a ν ∈Pη(M1) with ν|n = s then there isa ν′ ∈Pη(M0) with ν′|n = s.Note that any L1-elementary substructure of M1 is amenable.

Moreover,it easy to see that i) each Dη is a perfect tree, ii) 2ω is a disjoint union ofthe Dη and iii) for each s ∈2<ω there are 2ω sequences η such that s has anextension b ∈Dη.2.3 Theorem. The theory FERω of countably many refining equivalencerelations with binary splitting has a pair of models of size the continuumwhich are not isomorphic but are potentially isomorphic.This result follows from the next two propositions and the fact that M1is not rigid.12

2.4 Proposition. There is an L1-elementary substructure M0 of M1 suchthati) |Pη(M1) −Pη(M0)| ≤1.ii) M0|L0 is rigid.Proof.

Note that each automorphism of M1|L is determined by its restric-tion to the eventually constant sequences so there are only 2ω such. Thus wemay let ⟨fi : i < 2ω⟩enumerate the nontrivial automorphisms of M1|L.

Wedefine by induction disjoint subsets Ai, Bi of M1 each with cardinality lessthan the continuum. We denote ∪i

At stage i, choose α ∈M1such that fi moves α. Then, by continuity, there is a finite sequence s suchthat every element of Ws = {τ : s ⊆τ} is moved by fi.

Since |Ai|, |Bi| < 2ωand by condition iii) of the remark after the definition of amenable there arean η ∈2ω and a β ∈Pη∩(fi(Ws)−Ai). Then let Bi = {β} and Ai = Sη−{β}.Finally, let M0 = M1 −B2ω.Since no element is ever removed from an Ai, condition i) is satisfied.It is easy to see that M0 is rigid, as any nontrivial automorphism h of M0would extend in a unique way to an automorphism fi of M1 but at step i weensured that the restriction of fi to M0 is not an automorphism.2.5 Proposition.

If M0 is an amenable substructure of M1, M0 and M1 arepotentially isomorphic.Proof. Let P be the collection of all finite partial L1-isomorphisms be-tween M0 and M1.We first claim that P is a ccc set of forcing conditions.

In fact, P =∪n∈ωFn where if p, q ∈Fn then p ∪q ∈P. Given p ∈P, fix an (arbitrary)enumeration ⟨a1 .

. .

an⟩of dom p. Let n(p) be the cardinality of dom p andlet k(p) be the least integer satisfying the following conditions.i) If i ̸= j, M0 |= ¬Ek(ai, aj).ii) If i ̸= j, M1 |= ¬Ek(p(ai), p(aj)).iii) For all n with 2n + 1 ≥k, ai(2n + 1) = 0.iv) For all n with 2n + 1 ≥k, p(ai)(2n + 1) = 0.13

Now define an equivalence relation on P by p ≃q if n(p) = n(q), k(p) =k(q) and letting ⟨a1 . .

. an⟩enumerate dom p, ⟨a′1 .

. .

a′n⟩enumerate dom q,for each i, ai|k(p) = a′i|k(p) and p(ai)|k(p) = q(a′i)|k(p). Then ≃has onlycountably many equivalence classes; these classes are the Fn.

We must showthat if p ≃q then p ∪q ∈P. It suffices to show that for all i, j if M0 |=En(ai, a′j) then M1 |= En(p(ai), q(a′j)).• Case 1: i ̸= j.

By the definition of k = k(p), M0 |= ¬Ek(ai, a′j). Let ℓbe maximal so that M0 |= Eℓ(ai, a′j).

Then ℓ≥n since M0 |= En(ai, a′j).Since aj|k = a′j|k, ℓis also maximal so that M0 |= Eℓ(aj, ai). As p isan isomorphism, M1 |= Eℓ(p(ai), p(a′j)).

But p(aj)|k = q(a′j)|k, so ℓis also maximal with M1 |= Eℓ(q(a′j), p(ai)). Since n ≤ℓwe concludeM1 |= En(q(a′j), p(ai)) as required.• Case 2: i = j.

Suppose ai ∈Sν and a′j ∈Sη. We have M0 |= Ek(ai, a′j)by the definition of k and similarly, M1 |= Ek(p(ai), q(a′i)).

Now weshow by induction that for each m ≥k, M0 |= Em(ai, a′i) if and onlyif M1 |= Em(p(ai), q(a′i)). Assuming this condition for m we show itfor m + 1.

If m + 1 is odd, the result is immediate by parts iii) andiv) of the conditions defining ≃. If m = 2u then ai(m) = ν(u) anda′i(m) = η(u).

Since p and q are L1-isomorphisms p(ai)(m) = ν(u) andq(a′i)(m) = η(u). But M0 |= Em(ai, a′i) implies ν(u) = η(u) so we haveM1 |= Em(p(ai), q(a′i)).To show that the generic object is a map with domain M0, it suffices toshow that for any p ∈P and any a ∈M0 −dom p there is a q ∈P with q ≤pand dom q = dom p ∪{a}.

Choose n so that the members of dom p ∪{a} arepairwise En-inequivalent. Fix any L1-automorphism g of M∗that extends p.Let s = g(a)|n and Ws = {γ ∈2ω : s ⊆γ}.

Since there is a ν ∈C ∩Ws andB is amenable, there is a ν′ ∈B ∩Ws. Choosing ν′ for b, p ∪⟨a, b⟩is therequired extension of p.Since M0 and M1 are isomorphic in a generic extension for this forcing,we complete the proof.2.6 Remark.

The notion of a classifiable theory having two non-isomorphic,potentially isomorphic models is not very robust, and in particular can be lostby adding constants. As an example, let FER∗ω be an expansion of FERω14

formed by adding constants for the elements of a given countable model ofFERω. Then every type in this expanded language is stationary and theisomorphism type of any model of FER∗ω is determined by the number ofrealizations of each of the 2ω non-algebraic 1-types.Thus, if two modelsof FER∗ω are non-isomorphic then they remain non-isomorphic under anycardinal-preserving forcing.Similarly, non-isomorphism of models of the theory CEFω of countablymany crosscutting equivalence relations (i.e., Th(2ω, Ei)i∈ω, where Ei(σ, τ)iffσ(i) = τ(i)) is preserved under ccc forcings.We next want to extend the result from Theorem 2.3 to a larger classof theories.

Suppose T is superstable and there is a type q, possibly over afinite set e of parameters, and an e-definable family {En : n ∈ω} of properlyrefining equivalence relations, each with finitely many classes that determinethe strong types extending q. Let T be such a theory in a language L andlet M be a model of T. Let L0 be a reduct of L containing the En’s.We say ⟨aη ∈M : η ∈X ⊆2ω⟩is a set of unordered tree L-indiscerniblesif the following holds for any two sequences η, ν from X:If η and ν realize the same L0-type then ⟨aη1, .

. .

aηn⟩and ⟨aν1, . .

. aνn⟩satisfy the same L-type.We say that a superstable theory T with a type of infinite multiplicityas above embeds an unordered tree if there is a model M of T containing aset of unordered tree L-indiscernibles indexed by 2ω.

We deduce below theexistence of potentially isomorphic nonisomorphic models of weakly minimaltheories which embed an unordered tree. Every small superstable, non-ω-stable theory has a type of infinite multiplicity with an associated family of{En : n < ω} of refining equivalence relations and a set of tree indiscerniblesin the sense of [1].The existence of such a tree of indiscernibles sufficesfor the many model arguments but does not in itself suffice for this result.Marker has constructed an example of such a theory which does not embedan unordered tree.However, an apparently ad hoc argument shows thisexample does have potentially isomorphic but not isomorphic models.2.7 Notation.

Given A = {aη : η ∈2ω} a set of unordered tree indis-cernibles let D = {aη ∈A : η(n) = 0 for all but finitely many n}. For η ∈2ωlet pη(x) ∈S1(D) be q(x) ∪{En(x, aν) : aν ∈D and ν|n = η|n}.

Note thatD is a dense subset of A, each aη realizes pη and each pη is stationary.15

2.8 Lemma. Let T be a weakly minimal theory that embeds an unorderedtree.

Fix A and D as described in Notation 2.7. There is a set X satisfyingthe following conditions:i) X ∪A is independent over the empty set;ii) for any Y with D ⊆Y ⊆A, and any η ∈2ω, pη is realized in acl(XY )if and only if pη is realized in Y ;iii) for any Y with D ⊆Y ⊆A, acl(XY ) is a model of T.Proof.

It is easy to see from the definition of unordered tree indiscernibilitythat if X = ∅, then conditions i) and ii) of the Lemma are satisfied for anyY ⊆A. We will show that for any X and Y with D ⊂Y ⊆A with XYsatisfying conditions i) and ii) and any consistent formula φ(v) over acl(XY )that is not satisfied in acl(XY ) it is possible to adjoin a solution of φ toX while preserving the conditions.

By iterating this procedure we obtain amodel of T.Now suppose there is a Y with D ⊆Y ⊆A, such that acl(XY ) is not anelementary submodel of the monster. Choose a formula φ(x, c, a) with c ∈Xand a ∈Y such that φ(x, c, a) has a solution d in M but not in acl(XY ).

Ifwe adjoin d to X we must check that conditions i) and ii) are not violated.Since T is weakly minimal and d ̸∈acl(XY ), XAd is independent. Supposefor contradiction that for some a′ ∈Y , pν is not realized in a′ but pν isrealized in acl(Xda′) by say e. Since condition ii) holds for XY , e ̸∈acl(Xa′).Therefore by the exchange lemma d ∈acl(Xea′).

Let θ(v, c′, a,′ e) with c′ ∈Xand a′ ∈Y witness this algebraicity. Thenχ(c, c′, a, a′, z) = (∃x)[φ(x, c, a) ∧θ(x, c′, a′, z)] ∧(∃=mx)θ(x, c′, a′, z)is a formula over Xaa′ satisfied by e.Moreover, e ̸∈acl(Xaa′).For, ifso, transitivity would give d ∈acl(Xaa′) ⊆acl(XY ).Now tp(e/Xaa′)and in particular χ(c, c′, a, a′, z) is implied by pν and the assertion thatz ̸∈acl(Xaa′).

Since XA is independent, it follows by compactness thatthere is b ∈D such that χ(c, c′, a, a′, b) holds.So there is a solution ofφ(x, c, a) in the algebraic closure of XY . This contradicts the original choiceof φ so we conclude that condition ii) cannot be violated.16

2.9 Theorem. If T is a weakly minimal theory in a language of cardinalityat most 2ℵ0 that embeds an unordered tree then T has two models that arenot isomorphic but are potentially isomorphic (by a ccc-forcing).Proof.

Let L be the language of T. Assume that the type q is based ona finite set e. Let T ′ be the expansion of T formed by adding constants fore. Let M be a large saturated model of the theory T ′ and let the sets A andD be chosen as in Lemma 2.8 applied in L′ to T ′.Recall the definition of C from Notation 2.1.

For any W ⊆C, let M′W bethe L′-structure with universe acl(X ∪{aη : η ∈W}) and denote M′W|L byMW. We will construct an amenable set W such that MW ̸≈MC.

Since bothare amenable, there is a forcing extension where W ≈C as L1-structures.Since {aη : η ∈C} is a set of unordered tree L′-indiscernibles, the inducedmapping of {aη : η ∈W} into {aη : η ∈C} is L′-elementary. Thus, M′W ≈L′M′C and a fortiori MW ≈L MC.To construct W, let {fη : η ∈2ω} enumerate all L-embeddings of De intoMC.

Note that each pη can be considered as a complete L-type over De.Let W = ∪η∈2ωS′η where S′η = Sη −{bη} if MC realizes fη(pbη) and S′η =Sη if MC omits fη(pbη). (see Notation 2.1.

)Suppose for contradiction that g is an L-isomorphism between MW andMC.Then for some η, g|D = fη.Now if cη ∈W, the definition of Wyields fη(pη) is not realized in MC.This contradicts the choice of g asan isomorphism.But if cη is not in W then by the construction of W,fη(cη) = g(cη) does not realize g(pη). But this is impossible since g is ahomomorphism.References[1] J.T.

Baldwin. Diverse classes.

Journal of Symbolic Logic, 54:875–893,1989. [2] Steve Buechler and Saharon Shelah.

On the existence of regular types.Annals of Pure and Applied Logic, 45:207–308, 1989. [3] Bradd Hart.

Some results in classification theory. PhD thesis, McGillUniversity, 1986.17

[4] S. Shelah. Classification Theory and the Number of Nonisomorphic Mod-els.

North-Holland, 1978. [5] S. Shelah.

Existence of many L∞,λ-equivalent non-isomorphic models ofT of power λ. Annals of Pure and Applied Logic, 34, 1987. [6] S. Shelah.

Universal classes: Part 1. In J. Baldwin, editor, ClassificationTheory, Chicago 1985, pages 264–419.

Springer-Verlag, 1987. SpringerLecture Notes 1292.

[7] S. Shelah. Classification Theory and the Number of Nonisomorphic Mod-els.

North-Holland, 1991. second edition.18

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