Using Set Theory in Model Theory Kilpisjärvi 2013

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Using Set theory in model theory Kilpisjarvi¨ 2013 John T. Baldwin Using Set theory in model theory Context Kilpisjarvi¨ 2013 Absoluteness of Existence

Set Theoretic Method John T. Baldwin Analytically Presented AEC

Almost Galois ω-stability and absoluteness February 2, 2013 of ℵ1- categoricity

Models in ℵ2 Today’s Topics

Using Set theory in model theory Kilpisjarvi¨ 2013 1 Context John T. Baldwin 2 Absoluteness of Existence Context Absoluteness 3 Set Theoretic Method of Existence

Set Theoretic Method 4 Analytically Presented AEC Analytically Presented AEC 5 Almost Galois ω-stability and absoluteness of Almost Galois ω-stability and ℵ1-categoricity absoluteness of ℵ1- categoricity 6 Models in ℵ2 Models in ℵ2 References

Using Set theory in model theory Kilpisjarvi¨ 2013 John T. Baldwin

Context New results are from papers by Baldwin/Larson Absoluteness of Existence and Set Theoretic Baldwin/Larson/Shelah Method that are on my website. Analytically Presented AEC

Almost Galois ω-stability and absoluteness of ℵ1- categoricity

Models in ℵ2 1 The result may guide intuition towards a ZFC result. 2 Perhaps the hypothesis is eliminable A The combinatorial hypothesis might be replaced by a more subtle argument. E.G. Ultrapowers of elementarily equivalent models are isomorphic B The conclusion might be absolute The elementary equivalence proved in the Ax-Kochen-Ershov theorem C Consistency may imply truth.

Using Extensions of ZFC in Model Theory

Using Set theory in model theory A theorem under additional hypotheses is better than no Kilpisjarvi¨ 2013 theorem at all. John T. Baldwin

Context

Absoluteness of Existence

Set Theoretic Method

Analytically Presented AEC

Almost Galois ω-stability and absoluteness of ℵ1- categoricity

Models in ℵ2 Using Extensions of ZFC in Model Theory

Using Set theory in model theory A theorem under additional hypotheses is better than no Kilpisjarvi¨ 2013 theorem at all. John T. Baldwin 1 The result may guide intuition towards a ZFC result. Context 2 Perhaps the hypothesis is eliminable Absoluteness of Existence A The combinatorial hypothesis might be replaced by a

Set Theoretic more subtle argument. Method E.G. Ultrapowers of elementarily equivalent models are Analytically isomorphic Presented AEC B The conclusion might be absolute Almost Galois ω-stability and absoluteness The elementary equivalence proved in the of ℵ1- Ax-Kochen-Ershov theorem categoricity

Models in ℵ2 C Consistency may imply truth. Sacks Dicta

Using Set theory in model theory Kilpisjarvi¨ 2013 John T. Baldwin “... the central notions of model theory are absolute and absoluteness, unlike cardinality, is a logical concept. That is Context why model theory does not founder on that rock of Absoluteness of Existence undecidability, the generalized continuum hypothesis, and Set Theoretic why the Łos conjecture is decidable.” Method Analytically Gerald Sacks, 1972 Presented AEC See also the Vaananen article in Model Theoretic Logic Almost Galois ω-stability and volume absoluteness of ℵ1- categoricity

Models in ℵ2 Which ‘Central Notions’?

Using Set theory in model theory Kilpisjarvi¨ 2013 Chang’s two cardinal theorem (morasses) John T. Baldwin ‘Vaughtian pair is absolute’

Context

Absoluteness saturation is not absolute of Existence

Set Theoretic Aside: For aec, saturation is absolute below a categoricity Method cardinal. Analytically Presented AEC ‘elementarily prime model’ is absolute. (countable and Almost Galois ω-stability and atomic) absoluteness of ℵ1- categoricity ‘algebraically prime model’ is open

Models in ℵ2 Study Theories not logics

Using Set theory in model theory Kilpisjarvi¨ 2013 John T. Baldwin The most fundamental of Shelah’s innovations was to shift Context the focus from properties of logics (completeness, Absoluteness of Existence preservation theorem, compactness) Set Theoretic to Method

Analytically Classifying theories in a model theoretically fruitful way. Presented AEC (stable not decidable)

Almost Galois ω-stability and absoluteness of ℵ1- categoricity

Models in ℵ2 Classiﬁcation Theory

Using Set theory in model theory Kilpisjarvi¨ 2013 Crucial Observation John T. Baldwin The stability classiﬁcation is absolute.

Context

Absoluteness Fundamental Consequence of Existence

Set Theoretic Crucial properties are provable in ZFC for certain classes of Method theories. Analytically Presented 1 All stable theories have full two cardinal transfer. AEC

Almost Galois 2 There are saturated models exactly in the cardinals ω-stability and absoluteness where the theory is stable. of ℵ1- categoricity

Models in ℵ2 But this is for FIRST ORDER theories. Geography

Using Set theory in model theory Kilpisjarvi¨ 2013 John T. ˆ Baldwin Lω,ω⊂Lω1,ω⊂Lω1,ω(Q)⊂anal.pres.AEC⊂AEC.

Context

Absoluteness In a central case explained below of Existence

Set Theoretic Extensions of ZFC are used for Lω1,ω. Method ˆ Analytically Lω1,ω(Q) means only negative occurences of Q: e.g. Zilber Presented AEC ﬁeld, counterexample to absoluteness

Almost Galois ω-stability and Extensions of ZFC are proved necessary for Lω1,ω(Q). absoluteness of ℵ1- categoricity

Models in ℵ2 Two notions of ‘use’

Using Set theory in model theory Kilpisjarvi¨ 2013 1 Some model theoretic results ‘use’ extensions of ZFC John T. Baldwin 2 Some model theoretic results are provable in ZFC,

Context using models of set theory.

Absoluteness of Existence This Talk Set Theoretic Method 1 Analytically A quick statement of some results of the ﬁrst kind Presented AEC 2 Discussion of several examples of the second method. Almost Galois 3 ω-stability and Some further problems whose ‘kind’ is unknown. absoluteness of ℵ1- 4 Some combinatorial problems that have arisen. categoricity

Models in ℵ2 One Completely General Result

Using Set theory in model theory Kilpisjarvi¨ 2013 John T. + Baldwin Theorem: (2λ < 2λ ) (Shelah) Context

Absoluteness Suppose λ ≥ LS(K ) and K is λ-categorical. For any of Existence Abstract Elementary class, if amalgamation fails in λ there Set Theoretic λ+ + Method are 2 models in K of cardinality λ .

Analytically Presented + AEC Is 2λ < 2λ needed? Almost Galois ω-stability and absoluteness of ℵ1- categoricity

Models in ℵ2 Is 2λ < 2λ+ needed?

Using Set theory in model theory Kilpisjarvi¨ 2013 John T. Baldwin 1 λ = ℵ0: Context a Deﬁnitely not provable in ZFC: There are Absoluteness L(Q)-axiomatizable examples of Existence

Set Theoretic i Shelah: many models with CH, ℵ1-categorical under MA Method ii Koerwien-Todorcevic: many models under MA, Analytically ℵ1-categorical from PFA. Presented AEC b Independence Open for Lω1,ω Almost Galois ω-stability and absoluteness of ℵ1- categoricity

Models in ℵ2 A simple Problem

Using Set theory in model theory Kilpisjarvi¨ 2013 John T. Baldwin

Context Let φ be a sentence of Lω1,ω. Absoluteness of Existence Question Set Theoretic Method Is the property φ has an uncountable model absolute? Analytically Presented AEC

Almost Galois ω-stability and absoluteness of ℵ1- categoricity

Models in ℵ2 False Start

Using Set theory in model theory Kilpisjarvi¨ 2013 John T. Baldwin Fact: Easy for complete sentences

Context If φ is a complete sentence in Lω1,ω, Absoluteness of Existence φ has an uncountable model if and only if there exist

Set Theoretic countable M ω1,ω N which satisfy φ. Method Analytically This property is Σ1 and done by Shoenﬁeld absoluteness. Presented 1 AEC Almost Galois Note: Lω ,ω satisﬁes downward Lowenheim-Skolem¨ for ω-stability and 1 absoluteness sentences but not for theories. of ℵ1- categoricity

Models in ℵ2 Fly in the ointment

Using Set theory in model theory Kilpisjarvi¨ 2013 There are uncountable models that have no John T. Lω1,ω-elementary submodel. Baldwin E.g. any uncountable model of the ﬁrst order theory of Context

Absoluteness inﬁnitely many independent unary predicates Pi . of Existence

Set Theoretic So the sentence saying every ﬁnite Boolean combination of Method the Pi is non-empty has an uncountable model and our Analytically Presented obvious criteria does not work. AEC

Almost Galois ω-stability and Note that if we add the requirement that each type is absoluteness realized at most once, then every model has cardinality of ℵ1- categoricity ≤ 2ℵ0 . Models in ℵ2 Old argument

Using Set theory in model theory Kilpisjarvi¨ 2013 John T. Baldwin A really basic proof Context

Absoluteness Karp (1964) had proved completeness theorems for Lω1,ω, of Existence and Keisler (late 60’s/ early 70’s) for Lω1,ω(Q), Set Theoretic Method Barwise-Kaufmann-Makkai for L(aa) Lω1,ω(aa). Analytically Presented But we look at another argument that introduces some AEC

Almost Galois useful concepts and methods. ω-stability and absoluteness of ℵ1- categoricity

Models in ℵ2 Larson’s characterization

Using Set theory in model theory Kilpisjarvi¨ 2013 John T. Baldwin Theorem

Context Given a sentence φ of Lω1,ω(aa)(τ), Absoluteness the existence of a τ-structure of size ℵ1 satisfying φ of Existence is equivalent to Set Theoretic ◦ Method the existence of a countable model of ZFC containing Analytically {φ} ∪ ω which thinks there is a τ-structure of size ℵ Presented 1 AEC satisfying φ. Almost Galois ω-stability and 1 absoluteness This property is Σ1 and done by Shoenﬁeld absoluteness. of ℵ1- categoricity

Models in ℵ2 Larger Cardinals

Using Set theory in model theory Kilpisjarvi¨ 2013 It is easy to see that there are sentences of L such that John T. ω1,ω Baldwin the existence of a model in ℵ2 depends on the continuum

Context hypothesis.

Absoluteness S. Friedman and M. Koerwien have shown. of Existence Set Theoretic Assume GCH (and large cardinals for independence of the Method Kurepa hypothesis) Analytically Presented 1 AEC For any α ∈ ω1 − {0, 1, ω} there is a sentence φα such

Almost Galois that the existence of a model in ℵα is not absolute. ω-stability and absoluteness 2 For ℵ3, there is a complete such sentence. of ℵ1- categoricity

Models in ℵ2 Method: ‘Consistency implies Truth’

Using Set theory in model theory Kilpisjarvi¨ 2013 John T. Baldwin Let φ be a τ-sentence in Lω1,ω(Q) such that it is consistent that φ has a model. Context Let A be the countable ω-model of set theory, containing φ, Absoluteness of Existence that thinks φ has an uncountable model. Set Theoretic Method Construct B, an uncountable model of set theory, which is Analytically an elementary extension of A Presented AEC such that B is correct about uncountability. Then the model Almost Galois ω-stability and of φ in B is actually an uncountable model of φ. absoluteness of ℵ1- categoricity

Models in ℵ2 How to build B

Using Set theory in model theory Kilpisjarvi¨ 2013 John T. Baldwin

Context MT Iterate a theorem of Keisler and Morley (reﬁned by Absoluteness of Existence Hutchinson). Set Theoretic ST Iterations of ‘special’ ultrapowers. Method Analytically ZFC◦denotes a sufﬁcient subtheory of ZFC for our purposes. Presented AEC

Almost Galois ω-stability and absoluteness of ℵ1- categoricity

Models in ℵ2 How to build B

Using Set theory in model theory Kilpisjarvi¨ 2013 John T. The main technical tool is the iterated generic elementary Baldwin embedding induced by the nonstationary ideal on ω1, which Context we will denote by NSω1 . Absoluteness of Existence The ultraﬁlter Set Theoretic Method M Forcing with the Boolean algebra (P(ω1)/NSω1 ) over a Analytically M Presented ZFC model M gives rise to an M-normal ultraﬁlter U on ω1 AEC M (i.e., every regressive function on ω1 in M is constant on a Almost Galois ω-stability and set in U). absoluteness of ℵ1- categoricity

Models in ℵ2 The Ultrapower

Using Set theory in model theory Kilpisjarvi¨ Given such M and U, we can form the generic ultrapower 2013 Ult(M, U), which consists of all functions in M with domain John T. Baldwin M ω1 , Context where for any two such functions f , g, and any relation R in Absoluteness of Existence {=, ∈}, fRg in Ult(M, U) if and only if M Set Theoretic {α < ω1 | f (α)Rg(α)} ∈ U. Method

Analytically Presented Nota Bene AEC Almost Galois If M is countable, Ult(M, U) is countable. ω-stability and absoluteness of ℵ1- By convention, we identify the well-founded part of the categoricity ultrapower Ult(M, U) with its Mostowski collapse. Models in ℵ2 One step in the construction

Using Set theory in ◦ model theory We use three crucial properties of ZFC . Kilpisjarvi¨ ◦ 2013 1 The theory ZFC holds in every structure of the form John T. Baldwin H(κ) or Vκ, where κ is a regular cardinal greater than ℵ 22 1 Context |P|ℵ1| Absoluteness 2 If P is a c.c.c. notion of forcing, 2 < θ and X is an of Existence elementary submodel of H(θ), then any forcing Set Theoretic ◦ Method extension of the transitive collapse of X satisﬁes ZFC . Analytically ◦ Presented 3 Whenever M is a model of ZFC and U is an AEC M M-ultraﬁlter on ω1 , M is elementarily embedded in Almost Galois ω-stability and Ult(M, U). absoluteness of ℵ1- 4 If U constructed as above, categoricity Ult(M, U) increases exactly the sets that M thinks are Models in ℵ 2 uncountable. Really distinct interations

Using Set theory in model theory Kilpisjarvi¨ Theorem (Larson) 2013 ◦ John T. If M is a countable model of ZFC + MAℵ1 and Baldwin

Context hMα, Gα, jα,γ : α ≤ γ ≤ ω1, i Absoluteness of Existence and Set Theoretic 0 0 0 Method hMα, Gα, jα,γ : α ≤ γ ≤ ω1, i Analytically Presented are two distinct iterations of M, then AEC M0 Almost Galois Mω1 ω1 ω-stability and P(ω) ∩ P(ω) ⊂ Mα, absoluteness of ℵ1- 0 categoricity where α is least such that Gα 6= Gα.

Models in ℵ2 Gα not deﬁned for α = ω1. The Model Theory

Using Set theory in model theory Kilpisjarvi¨ ZFC-Theorem: (Keisler, new proof Larson) 2013 {ola2thrm} John T. Baldwin Let F be a countable fragment of Lω1,ω(aa). If there exists a model of cardinality ℵ1 realizing uncountably many F-types, Context there exists a 2ℵ1 -sized family of such models, each of Absoluteness of Existence cardinality ℵ1 and pairwise realizing just countably many Set Theoretic Method F-types in common.

Analytically Presented AEC (ZFC +CH)-Corollary (Shelah)

Almost Galois ℵ1 ω-stability and If a sentence in Lω1,ω has less that 2 models in ℵ1 then it absoluteness of ℵ1- is (syntactically) ω-stable. categoricity

Models in ℵ2 CH used twice. Sketching New Proof:

Using Set Let F be the smallest fragment of L (Q) containing φ. theory in ω1,ω model theory Kilpisjarvi¨ 1 Choose θ so that H(θ) contains the model N and θ is 2013 large enough for preservation of ZFC◦ . John T. Baldwin 2 Choose X a countable elementary submodel of H(θ). 3 Let M be the transitive collapse of X, and let N be the Context 0

Absoluteness image of N under this collapse. of Existence 0 4 Let M be a c.c.c. forcing extension of M satisfying Set Theoretic Method Martin’s Axiom. to get really distinct ultrapowers. Analytically 0 Presented 5 Build a tree of generic ultrapower iterates of M giving AEC rise to 2ℵ1 many distinct iterations of M0, each of length Almost Galois ω-stability and ω1. absoluteness of ℵ1- 6 Since F-types can be coded by reals using an categoricity enumeration of F in M, the images of N0 under these Models in ℵ2 iterations will pairwise realize just countably many F-types in common. 2 Closure under direct limits of ≺K -chains; 3 Downward Lowenheim-Skolem.¨

ABSTRACT ELEMENTARY CLASSES

Analytically Presented AEC

Almost Galois ω-stability and absoluteness Examples of ℵ1- categoricity First order and Lω1,ω-classes Models in ℵ2 L(Q) classes have Lowenheim-Skolem¨ number ℵ1. 3 Downward Lowenheim-Skolem.¨

ABSTRACT ELEMENTARY CLASSES

Almost Galois ω-stability and absoluteness Examples of ℵ1- categoricity First order and Lω1,ω-classes Models in ℵ2 L(Q) classes have Lowenheim-Skolem¨ number ℵ1. ABSTRACT ELEMENTARY CLASSES

Using Set theory in Generalizing Bjarni Jonsson:´ model theory Kilpisjarvi¨ 2013 ( , ≺ ) John T. A class of L-structures, K K , is said to be an abstract Baldwin elementary class: AEC if both K and the binary relation ≺K Context are closed under isomorphism plus: Absoluteness of Existence 1 If A, B, C ∈ K , A ≺K C, B ≺K C and A ⊆ B then Set Theoretic A ≺ B; Method K Analytically 2 Closure under direct limits of ≺K -chains; Presented AEC 3 Downward Lowenheim-Skolem.¨ Almost Galois ω-stability and absoluteness Examples of ℵ1- categoricity First order and Lω1,ω-classes Models in ℵ2 L(Q) classes have Lowenheim-Skolem¨ number ℵ1. Chains of models in ℵ1

Using Set theory in model theory Let K be an aec with amalgamation and joint embedding Kilpisjarvi¨ 2013 property and Lowenheim-Skolem number ℵ0. Lessmann John T. Baldwin shows that if there exist a pair of countable limit models (M ≺ N) and a Galois type over a submodel of M that is Context K omitted in N, then there is such a pair of cardinality ℵ1. The Absoluteness of Existence crux is that all countable ordinals have coﬁnality ω. Set Theoretic Here all models have cardinality µ. Method

Analytically Presented Deﬁnition AEC A (µ, θ) limit model M is a continuous union = S Almost Galois M i∈θ Mi ω-stability and where i+1 is µ-universal over i of size µ. absoluteness M M of ℵ1- categoricity We say M is a limit model over N if there is such a Models in ℵ2 sequence of Mi where M0 = N. k-domination

Using Set theory in model theory Kilpisjarvi¨ 2013 Deﬁnition John T. Baldwin Given two limit sequences, (Mi )i<α, (Ni )i<β we say that Context ( i )i<β k-dominates ( i )i<α if for all i < α there exists Absoluteness N M of Existence k < ω such that Ni+k is a proper universal extension of Set Theoretic and S is a proper extension of S . We Method Mi i<ω Ni i<ω Mi ( )

Models in ℵ2 A combinatorial question

Using Set theory in model theory Kilpisjarvi¨ 2013 John T. Show‘domination’ is a property of the ﬁltrations, not the Baldwin models. Context Problem Absoluteness of Existence Find an example of two limit models M and N and respective Set Theoretic Method limit sequences hMi : i < αi; hNi : i < βi such that Analytically k (Mi )i<α

Models in ℵ2 Analytically Presented AEC

Using Set theory in model theory Kilpisjarvi¨ 2013 John T. Baldwin

Context Deﬁnition {apres} Absoluteness An abstract elementary class K with Lowenheim¨ number ℵ of Existence 0

Set Theoretic is analytically presented if the set of countable models in K, Method and the corresponding strong submodel relation ≺K, are Analytically Presented both analytic. AEC

Almost Galois ω-stability and absoluteness of ℵ1- categoricity

Models in ℵ2 Deﬁnition

Using Set theory in model theory Kilpisjarvi¨ 2013 Fact John T. Baldwin An AEC K is PCΓ(ℵ0, ℵ0)-presented: Context if the models are reducts of models a countable ﬁrst order Absoluteness of Existence theory in an expanded vocabulary which omit a countable Set Theoretic family of types Method and the submodel relation is given in the same way. Analytically Presented AEC AKA: Almost Galois ω-stability and 1 Keisler: PCδ over Lω ,ω absoluteness 1 of ℵ1- categoricity 2 Shelah: PC(ℵ0, ℵ0), ℵ0-presented

Models in ℵ2 More Precisely

Using Set theory in model theory Kilpisjarvi¨ Theorem 2013 John T. If K is AEC then K can be analytically presented iff and only Baldwin if its restriction to ℵ0 is the restriction to ℵ0 of a Context PCΓ(ℵ0, ℵ0)-AEC. Absoluteness of Existence The following is basically folklore. Set Theoretic Method Countable case Analytically Presented AEC The countable τ-models of an analytically presented class

Almost Galois can be represented as reducts to τ of a sentence in ω-stability and 0 0 absoluteness Lω1,ω(τ ) for appropriate τ ⊇ τ. of ℵ1- categoricity Moreover the class of countable pairs (M, N) such that

Models in ℵ ≺ Γ(ℵ , ℵ ) 2 M K N is also a PC 0 0 -class. Uncountable Case

Using Set theory in model theory Kilpisjarvi¨ 2013 John T. Baldwin Theorem {uncountpcganal}

Context All τ-models of an analytically presented AEC K can be ∗ ∗ Absoluteness represented as reducts to τ of a sentence θ in Lω1,ω(τ ) for of Existence appropriate τ ∗ ⊇ τ. Set Theoretic Method Moreover the class of pairs (M, N) such that M ≺K N is the Analytically class of reducts to τ 0 of models of θ∗. Presented AEC

Almost Galois Proof combines the countable case with the idea of the ω-stability and absoluteness proof of the presentation theorem. of ℵ1- categoricity

Models in ℵ2 Example

Using Set theory in model theory Kilpisjarvi¨ 2013 Groupable partial orders (Jarden varying Shelah) John T. Baldwin Let (K , ≺) be the class of partially ordered sets such that each connected component is a countable 1-transitive linear Context order (equivalently admits a group structure) Absoluteness of Existence with M ≺ N if M ⊆ N and no component is extended. Set Theoretic Method This AEC is analytically presented. Analytically Presented Add a binary function and say it is a group on each AEC component. Almost Galois ℵ1 ℵ0 ω-stability and But it has 2 models in ℵ1 and 2 models in ℵ0. absoluteness of ℵ1- categoricity Recall: this ‘is’ the pseudo-elementary counterexample to

Models in ℵ2 Vaught’ s conjecture. Galois Types

Using Set theory in model theory Kilpisjarvi¨ 2013 Let M ≺K N0, M ≺K N1, a0 ∈ N0 and a1 ∈ N1 realize the John T. Baldwin same Galois Type over M iff there exist a structure N ∈ K and strong embeddings Context f0 : N0 → N and f1 : N1 → N such that f0|M = f1|M and Absoluteness of Existence f0(a0) = f1(a1). Set Theoretic Method Realizing the same Galois type (over countable models) is Analytically an equivalence relation Presented AEC

Almost Galois ω-stability and EM absoluteness of ℵ1- if K satisﬁes the amalgamation property. categoricity ℵ0

Models in ℵ2 The Monster Model

Using Set theory in model theory Kilpisjarvi¨ 2013 If an Abstract Elementary Class has the amalgamation John T. Baldwin property and the joint embedding property for models of cardinality at most ℵ0 Context

Absoluteness and has at most ℵ1-Galois types over models of cardinality of Existence ≤ ℵ Set Theoretic 0 Method Analytically then there is an ℵ -monster model for K and Presented 1 M AEC the Galois type of a over a countable M is the orbit of a Almost Galois under the automorphisms of which ﬁx M. ω-stability and M absoluteness of ℵ1- So EM is an equivalence relation on . categoricity M

Models in ℵ2 Another Combinatorial Question

Using Set theory in model theory Kilpisjarvi¨ 2013 John T. Question Baldwin Let K be an aec in vocabulary τ of cardinality ℵ1 with Context amalgamation and joint embedding property and Absoluteness of Existence Lowenheim-Skolem number ℵ1. Set Theoretic Method Suppose there exist a pair of limit models (M ≺K N) in ℵ1 Analytically and a Galois type over a submodel of M that is omitted in N. Presented AEC Must there be a model of cardinality ℵ2 omitting p. Can one Almost Galois ﬁnd a counterexample using some known combinatorial ω-stability and absoluteness objects? of ℵ1- categoricity

Models in ℵ2 Some stability notions

Using Set theory in model theory Kilpisjarvi¨ 2013 John T. Baldwin Deﬁnition

Context 1 The abstract elementary class (K, ≺) is said to be Absoluteness of Existence Galois ω-stable if for each countable M ∈ K, EM has Set Theoretic countably many equivalence classes. Method

Analytically 2 The abstract elementary class (K, ≺) is Presented AEC almost Galois ω-stable if for each countable M ∈ K, no

Almost Galois EM has a perfect set of equivalence classes. ω-stability and absoluteness of ℵ1- categoricity

Models in ℵ2 Almost Galois Stable

Using Set theory in model theory Kilpisjarvi¨ 2013 John T. Baldwin

Context Absoluteness Well-orders of type at most ℵ under end-extension are an of Existence 1

Set Theoretic AEC where countable models have only ℵ1 Galois types. Method

Analytically Presented AEC

Almost Galois ω-stability and absoluteness of ℵ1- categoricity

Models in ℵ2 1 Galois equivalence is Σ1

Using Set theory in model theory Kilpisjarvi¨ On an analytically presented AEC, having the same Galois 2013 type over M is an analytic equivalence relation, EM . So by John T. Baldwin Burgess’s theorem we have the following trichotomy.

Context Theorem Absoluteness of Existence An analytically presented abstract elementary class (K, ≺) Set Theoretic is Method

Analytically 1 Galois ω-stable or Presented AEC 2 almost Galois ω-stable or Almost Galois ω-stability and 3 has a perfect set of Galois types over some countable absoluteness of ℵ1- model categoricity

Models in ℵ2 Again basically folklore. Keisler for Galois types

Using Set theory in model theory Theorem: (B/Larson) Kilpisjarvi¨ 2013 Suppose that John T. Baldwin 1 K is an analytically presented abstract elementary Context class; Absoluteness of Existence 2 N is a K-structure of cardinality ℵ1, and N0 is a Set Theoretic countable structure with N0 ≺K N; Method ω 3 P is a perfect set of E -inequivalent members of ω ; Analytically N0 {manyGtypes} Presented AEC 4 N realizes the Galois types of uncountably many Almost Galois ω-stability and members of P over N0. absoluteness ℵ1 of ℵ1- Then there exists a family of 2 many K-structures of categoricity cardinality ℵ1, each containing N0 and pairwise realizing just Models in ℵ 2 countably many P-classes in common. Lω1,ω-case

Using Set theory in model theory Kilpisjarvi¨ 2013 John T. Baldwin Fact: Hyttinen-Kesala, Kueker Context {ass} Absoluteness If a sentence in Lω ,ω, satisfying amalgamation and joint of Existence 1 embedding, is almost Galois ω-stable Set Theoretic Method then it is Galois ω-stable. Analytically Presented AEC What about analytically presented? Almost Galois ω-stability and absoluteness of ℵ1- categoricity

Models in ℵ2 Analytically presented Strictly Almost Galois ω-stable example

Using Set theory in model theory Kilpisjarvi¨ 2013 John T. Baldwin The ‘groupable partial order’ is almost Galois stable

Context Let (K , ≺) be the class of partially ordered sets such that Absoluteness each connected component is a countable 1-transitive linear of Existence

Set Theoretic order with M ≺ N if M ⊆ N and no component is extended. Method

Analytically Since there are only ℵ1-isomorphism types of components Presented ω AEC this class is almost Galois -stable.

Almost Galois ω-stability and This AEC is analytically presented. absoluteness of ℵ1- categoricity

Models in ℵ2 Smallness

Using Set theory in model theory Kilpisjarvi¨ 2013 John T. Baldwin Deﬁnition

Context ∗ ∗ 1 A τ-structure M is L -small for L a countable fragment Absoluteness of Existence of Lω1,ω(τ) if M realizes only countably many ∗ ∗ Set Theoretic L (τ)-types (i.e. only countably many L (τ)-n-types for Method each n < ω). Analytically Presented 2 τ AEC A -structure M is called small or Lω1,ω-small if M

Almost Galois realizes only countably many Lω1,ω(τ)-types. ω-stability and absoluteness of ℵ1- categoricity

Models in ℵ2 Why Smallness matters

Using Set theory in model theory Kilpisjarvi¨ 2013 John T. Baldwin

Context Fact Absoluteness {Scottsent} of Existence Each small model satisﬁes a Scott-sentence, a complete Set Theoretic Method sentence of Lω1,ω. Analytically Presented AEC

Almost Galois ω-stability and absoluteness of ℵ1- categoricity

Models in ℵ2 Getting small models I

Using Set theory in model theory Kilpisjarvi¨ 2013 John T. Baldwin

Context Theorem: Shelah {getsmall} Absoluteness of Existence If K is analytically presented and some model of cardinality ∗ ∗ Set Theoretic ℵ1 is L -small for every countable τ-fragment L of Lω1,ω, Method 0 Analytically then K has an Lω1,ω(τ)-small model M of cardinality ℵ1. Presented AEC

Almost Galois ω-stability and absoluteness of ℵ1- categoricity

Models in ℵ2 Getting small models II

Using Set theory in model theory Theorem: Baldwin/Shelah/Larson Kilpisjarvi¨ 2013 If K has a model in ℵ1 that is not Lω1,ω(τ)-small then John T. Baldwin 1 there are at least ℵ1 complete sentences of Lω1,ω(τ) which are satisﬁed by uncountable models in K ; Context

Absoluteness 2 K has uncountably many models in ℵ1; of Existence 3 K has uncountably many extendible models in ℵ . Set Theoretic 0 Method Analytically Proof: Iterate the previous theorem. Presented AEC

Almost Galois Corollary: Baldwin/Shelah/Larson ω-stability and absoluteness Vaught’s conjecture is equivalent to Vaught’s conjecture for of ℵ1- categoricity extendible models.

Models in ℵ2

A countable model is extendible if it has an Lω1,ω-elementary extension. Complexity of (almost) ω-stability

Using Set theory in model theory Kilpisjarvi¨ 2013 John T. Baldwin 1 1 ﬁrst order syntactic: Π1 Context 1 2 Lω ,ω-syntactic: Π Absoluteness 1 1 of Existence 3 analytically presented AEC: Galois ω-stable: perhaps Set Theoretic 1 Method boldface Π4 Analytically 4 analytically presented AEC: almost Galois ω-stable: Presented AEC 1 boldface Π2 Almost Galois ω-stability and absoluteness of ℵ1- categoricity

Models in ℵ2 Absoluteness of ℵ1-categoricity

Using Set theory in model theory Kilpisjarvi¨ 1 ℵ1-categoricity of a class K deﬁned in Lω1,ω is absolute 2013 between models of set theory that satisfy any one of John T. Baldwin the following conditions.

Context 1 K is ω-stable; Absoluteness 2 K has arbitrarily large members and K has of Existence amalgamation in ℵ0; Set Theoretic ℵ0 ℵ1 Method 3 2 < 2 .

Analytically http://homepages.math.uic.edu/˜jbaldwin/ Presented pub/singsep2010.pdf AEC

Almost Galois 2 ℵ1-categoricity of an analytically presented AEC K is ω-stability and absoluteness absolute between models of set theory in which of ℵ1- categoricity K is almost Galois ω-stable, satisﬁes amalgamation in Models in ℵ2 ℵ0, and has an uncountable model. Why is this absoluteness of ℵ1-categoricity true for AEC?

Using Set theory in model theory Kilpisjarvi¨ Fact 2013 John T. Suppose that K is an analytically presented AEC. Then the Baldwin following statements are equivalent. Context 1 There exist a countable M ∈ K and an N ∈ K of Absoluteness cardinality ℵ1 such that: of Existence M ≺K N; Set Theoretic Method the set of Galois types over M realized in N is Analytically countable; Presented AEC some Galois type over M is not realized in N. ◦ Almost Galois 2 There is a countable model of ZFC whose ω1 is ω-stability and absoluteness well-founded and which contains trees on ω giving rise of ℵ1- categoricity to K, ≺K and the associated relation ∼0, and satisﬁes Models in ℵ2 statement 1. Models in ℵ2

Using Set theory in model theory Harrington (slides by Marker) http://homepages.math. Kilpisjarvi¨ 2013 uic.edu/˜marker/harrington-vaught.pdf John T. and Larson have shown that a counterexample, φ ∈ L ,to Baldwin A Vaught’s conjecture has (models in ℵ1) of arbitrarily large Context Scott rank (as sentences in Lω2,ω. Absoluteness of Existence The (unrealized) goal is to construct a model in ℵ2 and thus Set Theoretic Method prove Vaught’s conjecture since Hjorth as shown if there is a Analytically c.e. there is one with no model in ℵ2. Presented AEC Problem Almost Galois ω-stability and absoluteness Find model theoretic properties of these models of high of ℵ1- categoricity Scott rank. E.G, can we show some subset of these models

Models in ℵ2 form an LA elementary chain? Summary

Using Set theory in model theory Kilpisjarvi¨ 2013 John T. 1 The set theoretic method provides a uniform method for Baldwin studying models of various inﬁnitary logics Context 2 We introduced analytically presented AEC and showed: Absoluteness of Existence i analytically presented = PCΓ(ℵ0, ℵ0) Set Theoretic ii Extended Keisler’s few models implies ω-stability Method theorem to this class Analytically Presented iii Assuming countably many models in ℵ1: AEC Almost Galois ω-stable implies Galois ω-stable Almost Galois ω-stability and iv ℵ1-categoricity absolute for Almost Galois ω-stable with absoluteness amalgamation. of ℵ1- categoricity

Models in ℵ2