ω-saturated

	ω-saturated In mathematical logic, and particularly in its subfield model theory, a saturated model ''M'' is one that realizes as many complete types as may be "reasonably expected" given its size. For example, an ultrapower model of the hyperreals is \aleph_1-saturated, meaning that every descending nested sequence of internal sets has a nonempty intersection. Definition Let ''κ'' be a finite or infinite cardinal number and ''M'' a model in some first-order language. Then ''M'' is called ''κ''-saturated if for all subsets ''A'' ⊆ ''M'' of cardinality less than ''κ'', the model ''M'' realizes all complete types over ''A''. The model ''M'' is called saturated if it is , ''M'', -saturated where , ''M'', denotes the cardinality of ''M''. That is, it realizes all complete types over sets of parameters of size less than , ''M'', . According to some authors, a model ''M'' is called countably saturated if it is \aleph_1-saturated; that is, it realizes all complete types over coun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Hyperreals In mathematics, hyperreal numbers are an extension of the real numbers to include certain classes of infinite and infinitesimal numbers. A hyperreal number x is said to be finite if, and only if, , x, for some integer $n$ . Similarly, $x$ is said to be infinitesimal if, and only if, $, x, <1/n$ for all positive integers $n$ . The term "hyper-real" was introduced by Edwin Hewitt in 1948. The hyperreal numbers satisfy the transfer principle , a rigorous version of Leibniz's heuristic [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Mathematical Logic Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and Mathematical analysis, analysis. In the early 20th century it was shaped by David Hilbert's Hilbert's program, program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Dense Linear Order In mathematics, a partial order or total order < on a set $X$ is said to be dense if, for all $x$ and $y$ in $X$ for which $x < y$ , there is a $z$ in $X$ such that $x < z < y$ . That is, for any two elements, one less than the other, there is another element between them. For total orders this can be simplified to "for any two distinct elements, there is another element between them", since all elements of a total order are comparable. Example The rational number s as a linearly ordered set are a densely ordered set in this sense, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Mathematical Logic Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and Mathematical analysis, analysis. In the early 20th century it was shaped by David Hilbert's Hilbert's program, program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Chen Chung Chang Chen Chung Chang () was a mathematician who worked in model theory. He obtained his PhD from Berkeley in 1955 on "Cardinal and Ordinal Factorization of Relation Types" under Alfred Tarski. He wrote the standard text on model theory. Chang's conjecture and Chang's model are named after him. He also proved the ordinal partition theorem (expressed in the arrow notation for Ramsey theory) ωω→(ωω,3)2, originally a problem of Erdős and Hajnal. He also introduced MV-algebras as models for Łukasiewicz logic. Chang was a professor at the mathematics department of the University of California, Los Angeles The University of California, Los Angeles (UCLA) is a public university, public Land-grant university, land-grant research university in Los Angeles, California, United States. Its academic roots were established in 1881 as a normal school the .... Selected publications * * * C. C. Chang. Algebraic analysis of many-valued logics. Transactions of the American Mathema ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Stable Theory 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Homogeneous Model 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]
	Elementary Embedding In model theory, a branch of mathematical logic, two structures ''M'' and ''N'' of the same signature ''σ'' are called elementarily equivalent if they satisfy the same first-order ''σ''-sentences. If ''N'' is a substructure of ''M'', one often needs a stronger condition. In this case ''N'' is called an elementary substructure of ''M'' if every first-order ''σ''-formula ''φ''(''a''1, …, ''a''''n'') with parameters ''a''1, …, ''a''''n'' from ''N'' is true in ''N'' if and only if it is true in ''M''. If ''N'' is an elementary substructure of ''M'', then ''M'' is called an elementary extension of ''N''. An embedding ''h'': ''N'' → ''M'' is called an elementary embedding of ''N'' into ''M'' if ''h''(''N'') is an elementary substructure of ''M''. A substructure ''N'' of ''M'' is elementary if and only if it passes the Tarski–Vaught test: every first-order formula ''φ''(''x'', ''b''1, …, ''b''''n'') with p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Prime Model In mathematics, and in particular model theory, a prime model is a model that is as simple as possible. Specifically, a model P is prime if it admits an elementary embedding into any model M to which it is elementarily equivalent (that is, into any model M satisfying the same complete theory as P). Cardinality In contrast with the notion of saturated model, prime models are restricted to very specific cardinalities by the Löwenheim–Skolem theorem. If L is a first-order language with cardinality \kappa and T is a complete theory over L, then this theorem guarantees a model for T of cardinality \max(\kappa,\aleph_0). Therefore, no prime model of T can have larger cardinality since at the very least it must be elementarily embedded in such a model. This still leaves much ambiguity in the actual cardinality. In the case of countable languages, all prime models are at most countably infinite. Relationship with saturated models There is a duality between the definitions of pri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Strongly Inaccessible In set theory, a cardinal number is a strongly inaccessible cardinal if it is uncountable, regular, and a strong limit cardinal. A cardinal is a weakly inaccessible cardinal if it is uncountable, regular, and a weak limit cardinal. Since about 1950, "inaccessible cardinal" has typically meant "strongly inaccessible cardinal" whereas before it has meant "weakly inaccessible cardinal". Weakly inaccessible cardinals were introduced by . Strongly inaccessible cardinals were introduced by and ; in the latter they were referred to along with \aleph_0 as ''Grenzzahlen'' ( English "limit numbers"). Every strongly inaccessible cardinal is a weakly inaccessible cardinal. The generalized continuum hypothesis implies that all weakly inaccessible cardinals are strongly inaccessible as well. The two notions of an inaccessible cardinal \kappa describe a cardinality \kappa which can not be obtained as the cardinality of a result of typical set-theoretic operations involving only sets of card ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Elementary Extension In model theory, a branch of mathematical logic, two structures ''M'' and ''N'' of the same signature ''σ'' are called elementarily equivalent if they satisfy the same first-order ''σ''-sentences. If ''N'' is a substructure of ''M'', one often needs a stronger condition. In this case ''N'' is called an elementary substructure of ''M'' if every first-order ''σ''-formula ''φ''(''a''1, …, ''a''''n'') with parameters ''a''1, …, ''a''''n'' from ''N'' is true in ''N'' if and only if it is true in ''M''. If ''N'' is an elementary substructure of ''M'', then ''M'' is called an elementary extension of ''N''. An embedding ''h'': ''N'' → ''M'' is called an elementary embedding of ''N'' into ''M'' if ''h''(''N'') is an elementary substructure of ''M''. A substructure ''N'' of ''M'' is elementary if and only if it passes the Tarski–Vaught test: every first-order formula ''φ''(''x'', ''b''1, …, ''b''''n'') with p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Back-and-forth Method In mathematical logic, especially set theory and model theory, the back-and-forth method is a method for showing isomorphism between countably infinite structures satisfying specified conditions. In particular it can be used to prove that * any two countably infinite densely ordered sets (i.e., linearly ordered in such a way that between any two members there is another) without endpoints are isomorphic. An isomorphism between linear orders is simply a strictly increasing bijection. This result implies, for example, that there exists a strictly increasing bijection between the set of all rational numbers and the set of all real algebraic numbers. * any two countably infinite atomless Boolean algebras are isomorphic to each other. * any two equivalent countable atomic models of a theory are isomorphic. * the Erdős–Rényi model of random graphs, when applied to countably infinite graphs, almost surely produces a unique graph, the Rado graph. * any two many-complete recursively ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]

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