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Abstract Elementary Class
In model theory, a discipline within mathematical logic, an abstract elementary class, or AEC for short, is a class of models with a partial order similar to the relation of an elementary substructure of an elementary class in first-order model theory. They were introduced by Saharon Shelah. Definition \langle K, \prec_K\rangle, for K a class of structures in some language L = L(K), is an AEC if it has the following properties: * \prec_K is a partial order on K. * If M\prec_K N then M is a substructure of N. * Isomorphisms: K is closed under isomorphisms, and if M,N,M',N'\in K, f\colon M\simeq M', g\colon N\simeq N', f\subseteq g, and M\prec_K N, then M'\prec_K N'. * Coherence: If M_1\prec_K M_3, M_2\prec_K M_3, and M_1\subseteq M_2, then M_1\prec_K M_2. * Tarski–Vaught chain axioms: If \gamma is an ordinal and \\subseteq K is a chain (i.e. \alpha<\beta<\gamma\implies M_\alpha\prec_K M_\beta), then: ** ** If |
Model Theory
In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (mathematical logic), mathematical structure), and their Structure (mathematical logic), models (those Structure (mathematical logic), structures in which the statements of the theory hold). The aspects investigated include the number and size of models of a theory, the relationship of different models to each other, and their interaction with the formal language itself. In particular, model theorists also investigate the sets that can be definable set, defined in a model of a theory, and the relationship of such definable sets to each other. As a separate discipline, model theory goes back to Alfred Tarski, who first used the term "Theory of Models" in publication in 1954. Since the 1970s, the subject has been shaped decisively by Saharon Shel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Superstable
In the mathematical field of model theory, a theory is called stable if it satisfies certain combinatorial restrictions on its complexity. Stable theories are rooted in the proof of Morley's categoricity theorem and were extensively studied as part of Saharon Shelah's classification theory, which showed a dichotomy that either the models of a theory admit a nice classification or the models are too numerous to have any hope of a reasonable classification. A first step of this program was showing that if a theory is not stable then its models are too numerous to classify. Stable theories were the predominant subject of pure model theory from the 1970s through the 1990s, so their study shaped modern model theory and there is a rich framework and set of tools to analyze them. A major direction in model theory is "neostability theory," which tries to generalize the concepts of stability theory to broader contexts, such as simple and NIP theories. Motivation and history A common goa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |