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17
Categoricity from one successor cardinal in Tame Abstract Elementary Classes
, 2005
"... We prove that from categoricity in λ + we can get categoricity in all cardinals ≥ λ + in a χtame abstract elementary classes which has arbitrarily large models and satisfies the amalgamation and joint embedding properties, provided λ> LS(K) and λ ≥ χ. For the missing case when λ = LS(K), we pro ..."
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We prove that from categoricity in λ + we can get categoricity in all cardinals ≥ λ + in a χtame abstract elementary classes which has arbitrarily large models and satisfies the amalgamation and joint embedding properties, provided λ> LS(K) and λ ≥ χ. For the missing case when λ = LS(K), we prove that K is totally categorical provided that K is categorical in LS(K) and LS(K) +.
Interpolation, Preservation, and Pebble Games
 Journal of Symbolic Logic
, 1996
"... Preservation and interpolation results are obtained for L1! and sublogics L ` L1! such that equivalence in L can be characterized by suitable backandforth conditions on sets of partial isomorphisms. 1 Introduction In the heyday of infinitary logic in the 1960's and 70's, most attention ..."
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Preservation and interpolation results are obtained for L1! and sublogics L ` L1! such that equivalence in L can be characterized by suitable backandforth conditions on sets of partial isomorphisms. 1 Introduction In the heyday of infinitary logic in the 1960's and 70's, most attention was focused on L!1! and its fragments (see e.g. Keisler [19]), since countable formulas seemed best behaved. The past decade has seen a renewed interest in L1! and its finite variable fragments L (k) 1! (for 2 k ! !) and the modal fragment L \Pi 1! (see e.g. Ebbinghaus and Flum [17] on the former and Barwise and Moss [9] on the latter), due to various connections with topics in computer science. These logics form a hierarchy of increasingly powerful logics L \Pi 1! ae L (2) 1! ae L (3) 1! ae : : : ae L (k) 1! ae : : : ae L1! ; with each of these inclusions being proper. Moreover, there is a useful and elegant algebraic characterization of equivalence in L in each of these logics L, from b...
EhrenfeuchtMostowski models in abstract elementary classes
 Logic and its Applications, Contemporary Mathematics
, 2005
"... We work in the context of an abstract elementary class (AEC) with the amalgamation and joint embedding properties and arbitrarily large models. We prove two results using EhrenfeuchtMostowski models: 1) Morley’s omitting types theorem – for Galois types. 2) If an AEC (with amalgamation) is categori ..."
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Cited by 9 (5 self)
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We work in the context of an abstract elementary class (AEC) with the amalgamation and joint embedding properties and arbitrarily large models. We prove two results using EhrenfeuchtMostowski models: 1) Morley’s omitting types theorem – for Galois types. 2) If an AEC (with amalgamation) is categorical in some uncountable power µ it is stable in (every) λ < µ. These results are lemmas towards Shelah’s consideration [12] of the downward transfer of categoricity, which we discuss in Section 6. This paper expounds some of the main ideas of [12], filling in vague allusions to earlier work and trying to separate those results which depend only on the EhrenfeuchtMostowski method from those which require more sophisticated stability theoretic tools. In [15], Shelah proclaims the aim of reconstructing model theory, ‘with no use of even traces compactness’. We analyze here one aspect of this program. Keisler organizes [8] around four kinds of constructions: the Henkin method, EhrenfeuchtMostowski models, unions of chains, and ultraproducts. The later history of model theory reveals a plethora of tools arising in stability theory. Fundamental is a notion of dependence which arises from Morley’s study of rank, and passes through various avatars of splitting, strong splitting, and dividing before being fully actualized in the first order setting as forking. We eschew this technique altogether in this paper– to isolate its role. The axioms of an AEC (K, � K), were first set down in [17]. We repeat for convenience. Definition 0.1. A class of Lstructures, (K, � K), is said to be an abstract elementary class: AEC if both K and the binary relation � K are closed under isomorphism and satisfy the following conditions. • A1. If M � K N then M ⊆ N. • A2. � K is a partial order on K. • A3. If 〈Ai: i < δ 〉 is � Kincreasing chain:
Notes on quasiminimality and excellence
 Bulletin of Symbolic Logic
"... Zilber’s proposes [60] to prove ‘canonicity results for pseudoanalytic ’ structures. Informally, ‘canonical means the theory of the structure in a suitable possibly infinitary language (see Section 2) has one model in each uncountable power ’ while ‘pseudoanalytic means the model of power 2 ℵ0 can ..."
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Zilber’s proposes [60] to prove ‘canonicity results for pseudoanalytic ’ structures. Informally, ‘canonical means the theory of the structure in a suitable possibly infinitary language (see Section 2) has one model in each uncountable power ’ while ‘pseudoanalytic means the model of power 2 ℵ0 can be taken as a reduct of an expansion of the complex numbers by analytic functions’. This program interacts with two other lines of research. First is the general study of categoricity theorems in infinitary languages. After initial results by Keisler, reported in [31], this line was taken up in a long series of works by Shelah. We place Zilber’s work in this context. The second direction stems from Hrushovski’s construction of a counterexample to Zilber’s conjecture that every strongly minimal set is ‘trivial’, ‘vector spacelike’, or ‘fieldlike’. This construction turns out to be very concrete example of the Abstract Elementary Classes which arose in Shelah’s analysis. This paper examines the intertwining of these three themes. The study of (C, +, ·, exp) leads one immediately to some extension of first order logic; the integers with all their arithmetic are first order definable in (C, +, ·, exp). Thus, the first order theory of complex exponentiation is horribly complicated; it is certainly unstable and so can’t be first order categorical. One solution is to use infinitary logic to pin down the pathology. Insist that the kernel of the exponential map is fixed as a single copy of the integers while allowing the rest of the structure to grow. We describe in Section 5 Zilber’s program to
ON THE NUMBER OF L∞ω1EQUIVALENT NONISOMORPHIC MODELS
"... Abstract. We prove that if ZF is consistent then ZFC + GCH is consistent with the following statement: There is for every k < ω a model of cardinality ℵ1 which is L∞ω1equivalent to exactly k nonisomorphic models of cardinality ℵ1. In order to get this result we introduce ladder systems and colo ..."
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Abstract. We prove that if ZF is consistent then ZFC + GCH is consistent with the following statement: There is for every k < ω a model of cardinality ℵ1 which is L∞ω1equivalent to exactly k nonisomorphic models of cardinality ℵ1. In order to get this result we introduce ladder systems and colourings different from the “standard ” counterparts, and prove the following purely combinatorial result: For each prime number p and positive integer m it is consistent with ZFC + GHC that there is a “good ” ladder system having exactly pm pairwise nonequivalent colourings. 1.
Tameness from large cardinal axioms
"... Abstract. We show that Shelah’s Eventual Categoricity Conjecture follows from the existence of class many strongly compact cardinals. This is the first time the consistency of this conjecture has been proven. We do so by showing that every AEC with LS(K) below a strongly compact cardinal κ is < κ ..."
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Abstract. We show that Shelah’s Eventual Categoricity Conjecture follows from the existence of class many strongly compact cardinals. This is the first time the consistency of this conjecture has been proven. We do so by showing that every AEC with LS(K) below a strongly compact cardinal κ is < κ tame and applying the categoricity transfer of Grossberg and VanDieren [GV06a]. These techniques also apply to measurable and weakly compact cardinals and we prove similar tameness results under those hypotheses. We isolate a dual property to tameness, called type shortness, and show that it follows similarly
Barwise: Infinitary Logic and Admissible Sets. The Bulletin of Symbolic Logic 10
, 2004
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A Hanf number for saturation and omission
"... Suppose t = (T, T1, p) is a triple of two countable theories T ⊆ T1 in vocabularies τ ⊂ τ1 and a τ1type p over the empty set. We show the Hanf number for the property: There is a model M1 of T1 which omits p, but M1 ↾ τ is saturated is essentially equal to the Löwenheim number of second order logic ..."
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Suppose t = (T, T1, p) is a triple of two countable theories T ⊆ T1 in vocabularies τ ⊂ τ1 and a τ1type p over the empty set. We show the Hanf number for the property: There is a model M1 of T1 which omits p, but M1 ↾ τ is saturated is essentially equal to the Löwenheim number of second order logic. In Section 4 we make exact computations of these Hanf numbers and note some interesting distinctions between ‘first order ’ and ‘second order quantification’. In particular, we show that if κ is uncountable, h 3 (Lω,ω(Q), κ) = h 3 (Lω1,ω, κ), where h3 is the ‘normal ’ notion of Hanf function (Definition 4.13.) Newelski asked in [New] whether it is possible to calculate the Hanf number of the following property PN. In a sense made precise in Theorem 0.2, we show the answer is no. In accordance with the original question, we focus on countable vocabularies for the first three sections. We deal with extensions to larger vocabularies in Section 4. Definition 0.1 We say M1  = t where t = (T, T1, p) is a triple of two theories in vocabularies τ ⊂ τ1, respectively, T ⊆ T1 and p is a τ1type over the empty set if M1
On inverse γsystems and the number of L∞,λequivalent, nonisomorphic models for λ singular
 Institute of Mathematics, The Hebrew University, Jerusalem, Israel and Rutgers University, Hill CtrBusch
"... 644 revision:19980709 modified:19980709 Suppose λ is a singular cardinal of uncountable cofinality κ. For a model M of cardinality λ, let No(M) denote the number of isomorphism types of models N of cardinality λ which are L∞λequivalent to M. In [She85] Shelah considered inverse κsystems A of a ..."
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644 revision:19980709 modified:19980709 Suppose λ is a singular cardinal of uncountable cofinality κ. For a model M of cardinality λ, let No(M) denote the number of isomorphism types of models N of cardinality λ which are L∞λequivalent to M. In [She85] Shelah considered inverse κsystems A of abelian groups and their certain kind of quotient limits Gr(A)/Fact(A). In particular Shelah proved in [She85, Fact 3.10] that for every cardinal µ there exists an inverse κsystem A such that A consists of abelian groups having cardinality at most µ κ and card ( Gr(A)/Fact(A) ) = µ. Later in [She86, Theorem 3.3] Shelah showed a strict connection between inverse κsystems and possible values of No (under the assumption that θκ < λ for every θ < λ): if A is an inverse κsystem of abelian groups having cardinality < λ, then there is a model M such that card(M) = λ and No(M) = card ( Gr(A)/Fact(A) ). The following was an immediate consequence (when θκ < λ for every θ < λ): for every nonzero µ < λ or µ = λκ there is a model Mµ of cardinality λ with No(Mµ) = µ. In this paper we show: for every nonzero µ ≤ λκ there is an inverse κsystem A of abelian groups having cardinality < λ such that card ( Gr(A)/Fact(A) ) = µ (under the assumptions 2κ < λ and θ <κ < λ for all θ < λ when µ> λ), with the obvious new consequence concerning the possible value of No. Specifically, the case No(M) = λ is possible when θκ < λ for every θ < λ. 1 1