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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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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:
Representing embeddability as set inclusion
 Journal of LMS (2nd series), No.185
, 1998
"... ABSTRACT. A few steps are made towards representation theory of embeddability among uncountable graphs. A monotone class of graphs is defined by forbidding countable subgraphs, related to the graph’s endstructure. Using a combinatorial theorem of Shelah it is proved: The complexity of the class in ..."
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Cited by 7 (2 self)
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ABSTRACT. A few steps are made towards representation theory of embeddability among uncountable graphs. A monotone class of graphs is defined by forbidding countable subgraphs, related to the graph’s endstructure. Using a combinatorial theorem of Shelah it is proved: The complexity of the class in every regular uncountable λ> ℵ1 is at least λ + + sup{µ ℵ0: µ < λ} For all regular uncountable λ> ℵ1 there are 2λ pairwise non embeddable graphs in the class having strong homogeneity properties. It is characterized when some invariants of a graph G ∈ Gλ have to be inherited by one of fewer than λ subgraphs whose union covers G. All three results are obtained as corollaries of a representation theorem (Theorem 1.10 below), that asserts the existence of a surjective homomorphism from the relation of embeddability over isomorphism types of regular cardinality λ> ℵ1 onto set inclusion over all subsets of reals or cardinality λ or less. Continuity properties of the homomorphism are used to extend the first result to all singular cardinals below the first cardinal fixed point of second order. The first result shows that, unlike what Shelah showed in the class of all graphs,
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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Cited by 5 (4 self)
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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
The Primal Framework II: Smoothness
, 2003
"... 360 revision:19890114 modified:19950904 This is the second in a series of articles developing abstract classification theory for classes that have a notion of prime models over independent pairs and over chains. It deals with the problem of smoothness and establishing the existence and uniquenes ..."
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360 revision:19890114 modified:19950904 This is the second in a series of articles developing abstract classification theory for classes that have a notion of prime models over independent pairs and over chains. It deals with the problem of smoothness and establishing the existence and uniqueness of a ‘monster model’. We work here with a predicate for a canonically prime model. In a forthcoming paper, entitled, ‘Abstract classes with few models have ‘homogeneousuniversal ’ models’, we show how to drop this predicate from the set of basic notions and still obtain results analogous to those here. Experience with both first order logic and more general cases has shown the advantages of working within a ‘monster ’ model that is both ‘homogeneousuniversal’ and ‘saturated’. Fraïssé [6] for the countable case and Jónsson [9] for arbitrary cardinalities gave algebraic conditions on a class K of models that guaranteed the existence of a model that is homogeneous and universal for K. Morley and Vaught [11] showed that if K is the class of models of a first order theory then the algebraic conditions of homogeneity and universality are equivalent to model theoretic conditions of saturation. First order stability theory works within the fiction of a monster model M. Such
DEPENDENT T AND EXISTENCE OF LIMIT MODELS
"... Abstract. Does the class of linear order have (one of the variants of) the so called (λ, κ)limit model? (necessarily unique), and naturally assuming some instances of G.C.H. We get some positive results. More generally, letting T be a complete first order theory and for simplicity assume G.C.H. For ..."
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Abstract. Does the class of linear order have (one of the variants of) the so called (λ, κ)limit model? (necessarily unique), and naturally assuming some instances of G.C.H. We get some positive results. More generally, letting T be a complete first order theory and for simplicity assume G.C.H. For regular λ> κ> T  does T have (variants of) a (λ, κ)limit models, except for stable T? For some, yes, the theory of dense linear order, for some, no. Moreover, for independent T we get negative results. We deal more with linear orders. [877] revision:[20090627] (comes from 20090627) Cantor proved that the rationals as a linear order is characterized up to isomorphism by being “a dense linear order with neither first nor last element which is countable, e.g. of cardinality ℵ0”. Hausdorff generalizes this. For transparency
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 2K4. Number 2. August I9K4 COUNTABLE HOMOGENEOUS TOURNAMENTS BY
"... Abstract. A tournament Tis called homogeneous just in case every isomorphism of subtournaments of smaller cardinality can be lifted to an automorphism of T. It is shown that there are precisely three homogeneous tournaments of power N(). Some analogous results for 2tournaments are obtained. In this ..."
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Abstract. A tournament Tis called homogeneous just in case every isomorphism of subtournaments of smaller cardinality can be lifted to an automorphism of T. It is shown that there are precisely three homogeneous tournaments of power N(). Some analogous results for 2tournaments are obtained. In this paper we characterize countable tournaments which are homogeneous in the sense of Fraissé [1]. As far as we know, interest in such a characterization dates from 1976 when Woodrow showed in his dissertation [10] that the only finite homogeneous tournaments have orders 1 and 3 and that up to isomorphism there are only two countably infinite homogeneous tournaments which do not embed the tournament D shown in Figure 1. The next year Woodrow and the author [7] characterized countable homogeneous graphs. This reawakened the author's interest in the tournament problem. Independently, Schmerl [9] characterized countable homogeneous partial orderings and also formulated the question as to which countable tournaments are homogeneous. Figure 1 Apart from that of Woodrow mentioned above, the only work relevant to the problem that we know of is the unpublished proof of R. Morrow that any homogeneous tournament embedding D must also embed every tournament of order 6. Here we shall show that a homogeneous tournament embedding D embeds every finite tournament. Our proof requires labeled tournaments. For n < « an «tournament is a tournament in which each vertex is labeled by exactly one of the natural numbers < n. As a byproduct of our characterization of countable homogeneous tournaments we also find a corresponding characterization for 2tournaments. The problem for «tournaments (n> 2) is open.
Recommended Text: Introduction to Model Theory
"... After a couple of weeks to introduce the fundamental concepts and set the context (material chosen from the first three chapters of the text), the course will proceed with the development ..."
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After a couple of weeks to introduce the fundamental concepts and set the context (material chosen from the first three chapters of the text), the course will proceed with the development