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Beta-model
In model theory, a mathematical discipline, a β-model (from the French "bon ordre", well-ordering) is a model which is correct about statements of the form "''X'' is well-ordered". The term was introduced by Mostowski (1959)K. R. Apt, W. Marek,Second-order Arithmetic and Some Related Topics (1973), p. 181 as a strengthening of the notion of ω-model. In contrast to the notation for set-theoretic properties named by ordinals, such as \xi-indescribability, the letter β here is only denotational. In set theory It's a consequence of Shoenfield's absoluteness theorem that the constructible universe L is a β-model. In analysis β-models appear in the study of the reverse mathematics Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining method can briefly be described as "going backwards from the theorems to the axioms", in cont ... of subsystems of second-order arithmetic. In this ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reverse Mathematics
Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining method can briefly be described as "going backwards from the theorems to the axioms", in contrast to the ordinary mathematical practice of deriving theorems from axioms. It can be conceptualized as sculpting out necessary conditions from sufficient ones. The reverse mathematics program was foreshadowed by results in set theory such as the classical theorem that the axiom of choice and Zorn's lemma are equivalent over ZF set theory. The goal of reverse mathematics, however, is to study possible axioms of ordinary theorems of mathematics rather than possible axioms for set theory. Reverse mathematics is usually carried out using subsystems of second-order arithmetic,Simpson, Stephen G. (2009), Subsystems of second-order arithmetic, Perspectives in Logic (2nd ed.), Cambridge University Press, doi:10.1017/CBO9780511581007, ISBN 978 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Indescribable Cardinal
In mathematics, a Q-indescribable cardinal is a certain kind of large cardinal number that is hard to axiomatize in some language ''Q''. There are many different types of indescribable cardinals corresponding to different choices of languages ''Q''. They were introduced by . A cardinal number κ is called Π-indescribable if for every Πm proposition φ, and set A ⊆ Vκ with (Vκ+n, ∈, A) ⊧ φ there exists an α < κ with (Vα+n, ∈, A ∩ Vα) ⊧ φ. Following Lévy, here one looks at formulas with m-1 alternations of quantifiers with the outermost quantifier being universal. Σ-indescribable cardinals are defined in a similar way. The idea is that κ cannot be distinguished (looking from below) from smaller cardinals by any formula of n+1-th order logic with m-1 alternations of quantifiers even with the advantage of an extra unary predicate symbol (for A). This implies that it is large because it means that there must be many smaller cardinals with similar prope ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absoluteness
In mathematical logic, a formula is said to be absolute to some class of structures (also called models), if it has the same truth value in each of the members of that class. One can also speak of absoluteness of a formula ''between'' two structures, if it is absolute to some class which contains both of them.. Theorems about absoluteness typically establish relationships between the absoluteness of formulas and their syntactic form. There are two weaker forms of partial absoluteness. If the truth of a formula in each substructure ''N'' of a structure ''M'' follows from its truth in ''M'', the formula is downward absolute. If the truth of a formula in a structure ''N'' implies its truth in each structure ''M'' extending ''N'', the formula is upward absolute. Issues of absoluteness are particularly important in set theory and model theory, fields where multiple structures are considered simultaneously. In model theory, several basic results and definitions are motivated by absol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constructible Universe
In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted by , is a particular class of sets that can be described entirely in terms of simpler sets. is the union of the constructible hierarchy . It was introduced by Kurt Gödel in his 1938 paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis". In this paper, he proved that the constructible universe is an inner model of ZF set theory (that is, of Zermelo–Fraenkel set theory with the axiom of choice excluded), and also that the axiom of choice and the generalized continuum hypothesis are true in the constructible universe. This shows that both propositions are consistent with the basic axioms of set theory, if ZF itself is consistent. Since many other theorems only hold in systems in which one or both of the propositions is true, their consistency is an important result. What is can be thought of as being built in "stages" resembling the constr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytical Hierarchy
In mathematical logic and descriptive set theory, the analytical hierarchy is an extension of the arithmetical hierarchy. The analytical hierarchy of formulas includes formulas in the language of second-order arithmetic, which can have quantifiers over both the set of natural numbers, \mathbb, and over functions from \mathbb to \mathbb. The analytical hierarchy of sets classifies sets by the formulas that can be used to define them; it is the lightface version of the projective hierarchy. The analytical hierarchy of formulas The notation \Sigma^1_0 = \Pi^1_0 = \Delta^1_0 indicates the class of formulas in the language of second-order arithmetic with number quantifiers but no set quantifiers. This language does not contain set parameters. The Greek letters here are lightface symbols, which indicate this choice of language. Each corresponding boldface symbol denotes the corresponding class of formulas in the extended language with a parameter for each real; see projective hierarch ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |