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 !-stability and absoluteness February 2, 2013 of @1- categoricity Models in @2 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 !-stability and absoluteness of Almost Galois !-stability and @1-categoricity absoluteness of @1- categoricity 6 Models in @2 Models in @2 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 !-stability and absoluteness of @1- categoricity Models in @2 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 !-stability and absoluteness of @1- categoricity Models in @2 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 !-stability and absoluteness The elementary equivalence proved in the of @1- Ax-Kochen-Ershov theorem categoricity Models in @2 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 !-stability and volume absoluteness of @1- categoricity Models in @2 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 !-stability and atomic) absoluteness of @1- categoricity ‘algebraically prime model’ is open Models in @2 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 !-stability and absoluteness of @1- categoricity Models in @2 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 !-stability and absoluteness where the theory is stable. of @1- categoricity Models in @2 But this is for FIRST ORDER theories. Geography Using Set theory in model theory Kilpisjarvi¨ 2013 John T. ^ Baldwin L!;!⊂L!1;!⊂L!1;!(Q)⊂anal:pres:AEC⊂AEC: Context Absoluteness In a central case explained below of Existence Set Theoretic Extensions of ZFC are used for L!1;!. Method ^ Analytically L!1;!(Q) means only negative occurences of Q: e.g. Zilber Presented AEC field, counterexample to absoluteness Almost Galois !-stability and Extensions of ZFC are proved necessary for L!1;!(Q). absoluteness of @1- categoricity Models in @2 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 !-stability and Some further problems whose ‘kind’ is unknown. absoluteness of @1- 4 Some combinatorial problems that have arisen. categoricity Models in @2 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 !-stability and absoluteness of @1- categoricity Models in @2 Is 2λ < 2λ+ needed? Using Set theory in model theory Kilpisjarvi¨ 2013 John T. Baldwin 1 λ = @0: Context a Definitely not provable in ZFC: There are Absoluteness L(Q)-axiomatizable examples of Existence Set Theoretic i Shelah: many models with CH, @1-categorical under MA Method ii Koerwien-Todorcevic: many models under MA, Analytically @1-categorical from PFA. Presented AEC b Independence Open for L!1;! Almost Galois !-stability and absoluteness of @1- categoricity Models in @2 A simple Problem Using Set theory in model theory Kilpisjarvi¨ 2013 John T. Baldwin Context Let φ be a sentence of L!1;!. Absoluteness of Existence Question Set Theoretic Method Is the property φ has an uncountable model absolute? Analytically Presented AEC Almost Galois !-stability and absoluteness of @1- categoricity Models in @2 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 L!1;!, Absoluteness of Existence φ has an uncountable model if and only if there exist Set Theoretic countable M !1;! N which satisfy φ. Method Analytically This property is Σ1 and done by Shoenfield absoluteness. Presented 1 AEC Almost Galois Note: L! ;! satisfies downward Lowenheim-Skolem¨ for !-stability and 1 absoluteness sentences but not for theories. of @1- categoricity Models in @2 Fly in the ointment Using Set theory in model theory Kilpisjarvi¨ 2013 There are uncountable models that have no John T. L!1;!-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 !-stability and Note that if we add the requirement that each type is absoluteness realized at most once, then every model has cardinality of @1- categoricity ≤ 2@0 . Models in @2 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 L!1;!, of Existence and Keisler (late 60’s/ early 70’s) for L!1;!(Q), Set Theoretic Method Barwise-Kaufmann-Makkai for L(aa) L!1;!(aa). Analytically Presented But we look at another argument that introduces some AEC Almost Galois useful concepts and methods. !-stability and absoluteness of @1- categoricity Models in @2 Larson’s characterization Using Set theory in model theory Kilpisjarvi¨ 2013 John T. Baldwin Theorem Context Given a sentence φ of L!1;!(aa)(τ), Absoluteness the existence of a τ-structure of size @1 satisfying φ of Existence is equivalent to Set Theoretic ◦ Method the existence of a countable model of ZFC containing Analytically fφg [ ! which thinks there is a τ-structure of size @ Presented 1 AEC satisfying φ. Almost Galois !-stability and 1 absoluteness This property is Σ1 and done by Shoenfield absoluteness. of @1- categoricity Models in @2 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. !1;! Baldwin the existence of a model in @2 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 2 !1 − f0; 1;!g there is a sentence φα such Almost Galois that the existence of a model in @α is not absolute. !-stability and absoluteness 2 For @3, there is a complete such sentence. of @1- categoricity Models in @2 Method: ‘Consistency implies Truth’ Using Set theory in model theory Kilpisjarvi¨ 2013 John T. Baldwin Let φ be a τ-sentence in L!1;!(Q) such that it is consistent that φ has a model. Context Let A be the countable !-model of set theory, containing φ, Absoluteness of Existence that thinks φ has an uncountable model.