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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 formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those 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 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 Shelah's stability theory. Compared to other areas of mathematical logic such as proof theory, model theory is often less concerned with formal rigour and closer in spirit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superstable
In the mathematical field of model theory, a complete theory is called stable if it does not have too many types. One goal of classification theory is to divide all complete theories into those whose models can be classified and those whose models are too complicated to classify, and to classify all models in the cases where this can be done. Roughly speaking, if a theory is not stable then its models are too complicated and numerous to classify, while if a theory is stable there might be some hope of classifying its models, especially if the theory is superstable or totally transcendental. Stability theory was started by , who introduced several of the fundamental concepts, such as totally transcendental theories and the Morley rank. Stable and superstable theories were first introduced by , who is responsible for much of the development of stability theory. The definitive reference for stability theory is , though it is notoriously hard even for experts to read, as mentioned, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strongly Compact Cardinal
In set theory, a branch of mathematics, a strongly compact cardinal is a certain kind of large cardinal. A cardinal κ is strongly compact if and only if every κ-complete filter can be extended to a κ-complete ultrafilter. Strongly compact cardinals were originally defined in terms of infinitary logic, where logical operators are allowed to take infinitely many operands. The logic on a regular cardinal κ is defined by requiring the number of operands for each operator to be less than κ; then κ is strongly compact if its logic satisfies an analog of the compactness property of finitary logic. Specifically, a statement which follows from some other collection of statements should also follow from some subcollection having cardinality less than κ. The property of strong compactness may be weakened by only requiring this compactness property to hold when the original collection of statements has cardinality below a certain cardinal λ; we may then refer to λ-compactness. A card ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Successor Cardinal
In set theory, one can define a successor operation on cardinal numbers in a similar way to the successor operation on the ordinal numbers. The cardinal successor coincides with the ordinal successor for finite cardinals, but in the infinite case they diverge because every infinite ordinal and its successor have the same cardinality (a bijection can be set up between the two by simply sending the last element of the successor to 0, 0 to 1, etc., and fixing ω and all the elements above; in the style of Hilbert's Hotel Infinity). Using the von Neumann cardinal assignment and the axiom of choice (AC), this successor operation is easy to define: for a cardinal number ''κ'' we have :\kappa^+ = \left, \inf \\ , where ON is the class of ordinals. That is, the successor cardinal is the cardinality of the least ordinal into which a set of the given cardinality can be mapped one-to-one, but which cannot be mapped one-to-one back into that set. That the set above is nonempty follows from ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Type (model Theory)
In model theory and related areas of mathematics, a type is an object that describes how a (real or possible) element or finite collection of elements in a mathematical structure might behave. More precisely, it is a set of first-order formulas in a language ''L'' with free variables ''x''1, ''x''2,…, ''x''''n'' that are true of a sequence of elements of an ''L''-structure \mathcal. Depending on the context, types can be complete or partial and they may use a fixed set of constants, ''A'', from the structure \mathcal. The question of which types represent actual elements of \mathcal leads to the ideas of saturated models and omitting types. Formal definition Consider a structure \mathcal for a language ''L''. Let ''M'' be the universe of the structure. For every ''A'' ⊆ ''M'', let ''L''(''A'') be the language obtained from ''L'' by adding a constant ''c''''a'' for every ''a'' ∈ ''A''. In other words, :L(A) = L \cup \. A 1-type (of \mathcal) over ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Continuum Hypothesis
In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that or equivalently, that In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to the following equation in aleph numbers: 2^=\aleph_1, or even shorter with beth numbers: \beth_1 = \aleph_1. The continuum hypothesis was advanced by Georg Cantor in 1878, and establishing its truth or falsehood is the first of Hilbert's 23 problems presented in 1900. The answer to this problem is independent of ZFC, so that either the continuum hypothesis or its negation can be added as an axiom to ZFC set theory, with the resulting theory being consistent if and only if ZFC is consistent. This independence was proved in 1963 by Paul Cohen, complementing earlier work by Kurt Gödel in 1940. The name of the hypothesis comes from the term '' the continuum'' for the real numbers. History Cantor believed the continuum hypothesis to be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Large Cardinals
In the mathematical field of set theory, a large cardinal property is a certain kind of property of Transfinite number, transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally very "large" (for example, bigger than the least α such that α=ωα). The proposition that such cardinals exist cannot be proved in the most common axiomatization of set theory, namely ZFC, and such propositions can be viewed as ways of measuring how "much", beyond ZFC, one needs to assume to be able to prove certain desired results. In other words, they can be seen, in Dana Scott's phrase, as quantifying the fact "that if you want more you have to assume more". There is a rough convention that results provable from ZFC alone may be stated without hypotheses, but that if the proof requires other assumptions (such as the existence of large cardinals), these should be stated. Whether this is simply a linguistic convention, or something more, is a controversial point am ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of '' naive set theory''. After the discovery of paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and the Burali-Forti paradox) various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is still the best-known and most studied. Set theory is commonly employed as a foundational ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morley's Categoricity Theorem
In mathematical logic, a theory is categorical if it has exactly one model (up to isomorphism). Such a theory can be viewed as ''defining'' its model, uniquely characterizing the model's structure. In first-order logic, only theories with a finite model can be categorical. Higher-order logic contains categorical theories with an infinite model. For example, the second-order Peano axioms are categorical, having a unique model whose domain is the set of natural numbers \mathbb. In model theory, the notion of a categorical theory is refined with respect to cardinality. A theory is -categorical (or categorical in ) if it has exactly one model of cardinality up to isomorphism. Morley's categoricity theorem is a theorem of stating that if a first-order theory in a countable language is categorical in some uncountable cardinality, then it is categorical in all uncountable cardinalities. extended Morley's theorem to uncountable languages: if the language has cardinality and a the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stable Theory
In the mathematical field of model theory, a complete theory is called stable if it does not have too many types. One goal of classification theory is to divide all complete theories into those whose models can be classified and those whose models are too complicated to classify, and to classify all models in the cases where this can be done. Roughly speaking, if a theory is not stable then its models are too complicated and numerous to classify, while if a theory is stable there might be some hope of classifying its models, especially if the theory is superstable or totally transcendental. Stability theory was started by , who introduced several of the fundamental concepts, such as totally transcendental theories and the Morley rank. Stable and superstable theories were first introduced by , who is responsible for much of the development of stability theory. The definitive reference for stability theory is , though it is notoriously hard even for experts to read, as mentioned, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tame Abstract Elementary Class
In model theory, a discipline within the field of mathematical logic, a tame abstract elementary class is an abstract elementary class (AEC) which satisfies a locality property for types called tameness. Even though it appears implicitly in earlier work of Shelah, tameness as a property of AEC was first isolated by Grossberg and VanDieren, who observed that tame AECs were much easier to handle than general AECs. Definition Let ''K'' be an AEC with joint embedding, amalgamation, and no maximal models. Just like in first-order model theory, this implies ''K'' has a universal model-homogeneous monster model \mathfrak. Working inside \mathfrak, we can define a semantic notion of types by specifying that two elements ''a'' and ''b'' have the same type over some base model M if there is an automorphism of the monster model sending ''a'' to ''b'' fixing M pointwise (note that types can be defined in a similar manner without using a monster model). Such types are called Galois types. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compactness Theorem
In mathematical logic, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model. This theorem is an important tool in model theory, as it provides a useful (but generally not effective) method for constructing models of any set of sentences that is finitely consistent. The compactness theorem for the propositional calculus is a consequence of Tychonoff's theorem (which says that the product of compact spaces is compact) applied to compact Stone spaces, hence the theorem's name. Likewise, it is analogous to the finite intersection property characterization of compactness in topological spaces: a collection of closed sets in a compact space has a non-empty intersection if every finite subcollection has a non-empty intersection. The compactness theorem is one of the two key properties, along with the downward Löwenheim–Skolem theorem, that is used in Lindström's theorem to characterize first-order logic. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
