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Using Set Theory in Model Theory ASL/AMS Annual Meeting Boston 2012 Using Set Theory in Model Theory John T Using Set theory in model theory ASL/AMS Annual Meeting Boston 2012 Using Set theory in model theory John T. Baldwin ASL/AMS Annual Meeting Boston 2012 John T. Baldwin January 8, 2012 Today’s Topics Using Set theory in model theory ASL/AMS Annual Meeting Boston 2012 John T. Baldwin 1 The result may guide intuition towards a ZFC result. 2 Perhaps the hypothesis is eliminable A The cominatorial 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 ASL/AMS Annual theorem at all. Meeting Boston 2012 John T. Baldwin 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 ASL/AMS Annual theorem at all. Meeting Boston 2012 John T. 1 The result may guide intuition towards a ZFC result. Baldwin 2 Perhaps the hypothesis is eliminable A The cominatorial hypothesis might be replaced by a more subtle argument. E.G. Ultrapowers of elementarily equivalent models are isomorphic 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 ASL/AMS Annual theorem at all. Meeting Boston 2012 John T. 1 The result may guide intuition towards a ZFC result. Baldwin 2 Perhaps the hypothesis is eliminable A The cominatorial 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 Using Extensions of ZFC in Model Theory Using Set theory in model theory A theorem under additional hypotheses is better than no ASL/AMS Annual theorem at all. Meeting Boston 2012 John T. 1 The result may guide intuition towards a ZFC result. Baldwin 2 Perhaps the hypothesis is eliminable A The cominatorial 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. Sacks Dicta Using Set theory in model theory ASL/AMS Annual Meeting Boston 2012 “... the central notions of model theory are absolute and John T. absoluteness, unlike cardinality, is a logical concept. That is Baldwin why model theory does not founder on that rock of undecidability, the generalized continuum hypothesis, and why the Łos conjecture is decidable.” Gerald Sacks, 1972 See also the Vaananen article in Model Theoretic Logic volume Which ‘Central Notions’? Using Set theory in model theory ASL/AMS Annual Meeting Boston 2012 John T. Chang’s two cardinal theorem (morasses) Baldwin ‘Vaughtian pair is absolute’ saturation is not absolute Aside: For aec, saturation is absolute below a categoricity cardinal. Classification Theory Using Set theory in model theory ASL/AMS Annual Crucial Observation Meeting Boston 2012 The stability classification is absolute. John T. Baldwin Fundamental Consequence Crucial properties are provable in ZFC for certain classes of theories. 1 All stable theories have full two cardinal transfer. 2 There are saturated models exactly in the cardinals where the theory is stable. But this is for FIRST ORDER theories. Geography Using Set theory in model theory ASL/AMS Annual Meeting Boston 2012 John T. Baldwin L!;!⊂L!1;!⊂L!1;!(Q)⊂anal:pres:AEC⊂AEC: In a central case explained below Extensions of ZFC are used for L!1;!. Extensions of ZFC are proved necessary for L!1;!(Q). Two notions of ‘use’ Using Set theory in model theory ASL/AMS Annual Meeting Boston 2012 1 Some model theoretic results ‘use’ extensions of ZFC John T. Baldwin 2 Some model theoretic results are provable in ZFC, using models of set theory. This Talk 1 A quick statement of some results of the first kind 2 Discussion of several examples of the second method. One Completely General Result Using Set theory in model theory ASL/AMS Annual Meeting Boston 2012 λ λ+ John T. Theorem: (2 < 2 ) (Shelah) Baldwin Suppose λ ≥ LS(K ) and K is λ-categorical. For any Abstract Elementary class, if amalgamation fails in λ there + are 2λ models in K of cardinality λ+. + Is 2λ < 2λ needed? Is 2λ < 2λ+ needed? Using Set theory in model theory ASL/AMS Annual Meeting Boston 2012 1 λ = @0: John T. a Definitely not provable in ZFC: There are Baldwin L(Q)-axiomatizable examples i Shelah: many models with CH, @1-categorical under MA ii Koerwien-Todorcevic: many models under MA, @1-categorical from PFA. b Independence Open for L!1;! 2 Grossberg and VanDieren have announced the AEC analog in larger λ using the generalized Martin’s Axiom. A simple Problem Using Set theory in model theory ASL/AMS Annual Meeting Boston 2012 John T. Baldwin Let φ be a sentence of L!1;!. Question Is the property φ has an uncountable model absolute? False Start Using Set theory in model theory ASL/AMS Annual Meeting Boston 2012 Fact: Easy for complete sentences John T. Baldwin If φ is a complete sentence in L!1;!, φ has an uncountable model if and only if there exist countable M !1;! N which satisfy φ. 1 This property is Σ1 and done by Shoenfield absoluteness. Note: L!1;! satisfies downward Lowenheim-Skolem¨ for sentences but not for theories. Fly in the ointment Using Set theory in model theory ASL/AMS Annual There are uncountable models that have no Meeting Boston 2012 L!1;!-elementary submodel. John T. Baldwin E.g. any uncountable model of the first order theory of infinitely many independent unary predicates Pi . So the sentence saying every finite Boolean combination of the Pi is non-empty has an uncountable model and our obvious criteria does not work. Note that if we add the requirement that each type is realized at most once, then every model has cardinality ≤ 2@0 . BUT NO! Asserted by Gregory; example found by Johnson, Knight, Ocasio, VanDenDriessche this Fall L∗-submodel Using Set theory in model theory ASL/AMS Annual Meeting ∗ Boston 2012 Given a sentence φ. Let L be the minimal countable John T. fragment of L! ;! containing φ. Baldwin 1 Suppose M ≺L∗ N, M 6= N. 0 0 Does there exist a proper extension N of N with N ≺L∗ N ? If so we have an absolute characterization of φ has a uncountable model. L∗-submodel Using Set theory in model theory ASL/AMS Annual Meeting ∗ Boston 2012 Given a sentence φ. Let L be the minimal countable John T. fragment of L! ;! containing φ. Baldwin 1 Suppose M ≺L∗ N, M 6= N. 0 0 Does there exist a proper extension N of N with N ≺L∗ N ? If so we have an absolute characterization of φ has a uncountable model. BUT NO! Asserted by Gregory; example found by Johnson, Knight, Ocasio, VanDenDriessche this Fall Smallness Using Set theory in model theory ASL/AMS Annual Meeting Boston 2012 Definition John T. Baldwin ∗ ∗ 1 A τ-structure M is L -small for L a countable fragment of L!1;!(τ) if M realizes only countably many L∗(τ)-types (i.e. only countably many L∗(τ)-n-types for each n < !). 2 A τ-structure M is called small or L!1;!-small if M realizes only countably many L!1;!(τ)-types. Why Smallness matters Using Set theory in model theory ASL/AMS Annual Meeting Boston 2012 John T. Baldwin Fact fScottsentg Each small model satisfies a Scott-sentence, a complete sentence of L!1;!. A Correct Characterization Using Set theory in model theory ASL/AMS Annual Meeting Boston 2012 Larson’s characterization John T. Baldwin Given a sentence φ of L!1;!(aa), the existence of a model of size @1 satisfying φ is equivalent to the existence of a countable model of ZFC◦ containing fφg [ ! which thinks there is a model of size @1 with a member satisfying φ. 1 This property is Σ1 and done by Shoenfield absoluteness. Larger Cardinals Using Set theory in model theory ASL/AMS Annual Meeting It is easy to see that there are sentences of L!1;! such that Boston 2012 the existence of a model in @ depends on the continuum John T. 2 Baldwin hypothesis. S. Friedman and M. Koerwien have shown. Assume GCH (and large cardinals for independence of the Kurepa hypothesis) 1 For any α 2 !1 − f0; 1;!g there is a sentence φα such that the existence of a model in @α is not absolute. 2 For @3, there is a complete such sentence. The rest of the talk illustrates the advantages of missing the ‘obvious’ argument. Deja vu Using Set theory in model theory ASL/AMS Annual Meeting Boston 2012 John T. Baldwin The really basic proof Karp (1964) had proved completeness theorems for L!1;!, and Keisler (late 60’s/ early 70’s) for L!1;!(Q), L!1;!(aa). Deja vu Using Set theory in model theory ASL/AMS Annual Meeting Boston 2012 John T. Baldwin The really basic proof Karp (1964) had proved completeness theorems for L!1;!, and Keisler (late 60’s/ early 70’s) for L!1;!(Q), L!1;!(aa). The rest of the talk illustrates the advantages of missing the ‘obvious’ argument.
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