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Minimal Model (set Theory)
In set theory, a branch of mathematics, the minimal model is the minimal standard model of ZFC. The minimal model was introduced by and rediscovered by . The existence of a minimal model cannot be proved in ZFC, even assuming that ZFC is consistent, but follows from the existence of a standard model as follows. If there is a ''set'' ''W'' in the von Neumann universe V that is a standard model of ZF, and the ordinal ''κ'' is the set of ordinals that occur in ''W'', then L''κ'' is the class of constructible sets of ''W''. If there is a set that is a standard model of ZF, then the smallest such set is such a L''κ''. This set is called the minimal model of ZFC, and also satisfies the axiom of constructibility V=L. The downward Löwenheim–Skolem theorem implies that the minimal model (if it exists as a set) is a countable set. More precisely, every element ''s'' of the minimal model can be named; in other words there is a first-order sentence ''φ''(''x'') such that ''s'' is the ... [...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] |
Standard Model (set Theory)
In set theory, a standard model for a theory T is a model M for T where the membership relation \in_M is the same as the membership relation \in of the set theoretical universe V (restricted to the domain of M). In other words, M is a substructure of V. A standard model M that satisfies the additional transitivity condition that x \in y \in M implies x \in M is a standard transitive model (or simply a transitive model). Usually, when one talks about a model M of set theory, it is assumed that M is a set model, i.e. the domain of M is a set in V. If the domain of M is a proper class, then M is a class model. An inner model In set theory, a branch of mathematical logic, an inner model for a theory ''T'' is a substructure of a model ''M'' of a set theory that is both a model for ''T'' and contains all the ordinals of ''M''. Definition Let L = \langle \in \rangle be ... is necessarily a class model. References * * {{uncategorized, date=April 2023 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Consistent
In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if it has a model, i.e., there exists an interpretation under which all formulas in the theory are true. This is the sense used in traditional Aristotelian logic, although in contemporary mathematical logic the term ''satisfiable'' is used instead. The syntactic definition states a theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences of T. Let A be a set of closed sentences (informally "axioms") and \langle A\rangle the set of closed sentences provable from A under some (specified, possibly implicitly) formal deductive system. The set of axioms A is consistent when \varphi, \lnot \varphi \in \langle A \rangle for no formula \varphi. If there ex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Von Neumann Universe
In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted by ''V'', is the class of hereditary well-founded sets. This collection, which is formalized by Zermelo–Fraenkel set theory (ZFC), is often used to provide an interpretation or motivation of the axioms of ZFC. The concept is named after John von Neumann, although it was first published by Ernst Zermelo in 1930. The rank of a well-founded set is defined inductively as the smallest ordinal number greater than the ranks of all members of the set. In particular, the rank of the empty set is zero, and every ordinal has a rank equal to itself. The sets in ''V'' are divided into the transfinite hierarchy ''Vα'', called the cumulative hierarchy, based on their rank. Definition The cumulative hierarchy is a collection of sets ''V''α indexed by the class of ordinal numbers; in particular, ''V''α is the set of all sets having ranks less than α. Thus there is one set ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inner Model
In set theory, a branch of mathematical logic, an inner model for a theory ''T'' is a substructure of a model ''M'' of a set theory that is both a model for ''T'' and contains all the ordinals of ''M''. Definition Let L = \langle \in \rangle be the language of set theory. Let ''S'' be a particular set theory, for example the ZFC axioms and let ''T'' (possibly the same as ''S'') also be a theory in L. If ''M'' is a model for ''S'', and ''N'' is an L-structure such that #''N'' is a substructure of ''M'', i.e. the interpretation \in_N of \in in ''N'' is \cap N^2 #''N'' is a model for ''T'' #the domain of ''N'' is a transitive class of ''M'' #''N'' contains all ordinals of ''M'' then we say that ''N'' is an inner model of ''T'' (in ''M''). Usually ''T'' will equal (or subsume) ''S'', so that ''N'' is a model for ''S'' 'inside' the model ''M'' of ''S''. If only conditions 1 and 2 hold, ''N'' is called a standard model of ''T'' (in ''M''), a standard submodel of ''T'' (if ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinal Number
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the least natural number that has not been previously used. To extend this process to various infinite sets, ordinal numbers are defined more generally as linearly ordered labels that include the natural numbers and have the property that every set of ordinals has a least element (this is needed for giving a meaning to "the least unused element"). This more general definition allows us to define an ordinal number \omega that is greater than every natural number, along with ordinal numbers \omega + 1, \omega + 2, etc., which are even greater than \omega. A linear order such that every subset has a least element is called a well-order. The axiom of choice implies that every set can be well-ordered, and given two well-ordered sets, one is isomorphic to ... [...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] |
Axiom Of Constructibility
The axiom of constructibility is a possible axiom for set theory in mathematics that asserts that every set is constructible universe, constructible. The axiom is usually written as ''V'' = ''L'', where ''V'' and ''L'' denote the von Neumann universe and the constructible universe, respectively. The axiom, first investigated by Kurt Gödel, is inconsistent with the proposition that zero sharp exists and stronger large cardinal axioms (see list of large cardinal properties). Generalizations of this axiom are explored in inner model theory. Implications The axiom of constructibility implies the axiom of choice (AC), given Zermelo–Fraenkel set theory without the axiom of choice (ZF). It also settles many natural mathematical questions that are independent of Zermelo–Fraenkel set theory with the axiom of choice (ZFC); for example, the axiom of constructibility implies the Continuum hypothesis#The generalized continuum hypothesis, generalized continuum hypothesis, the negation of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Löwenheim–Skolem Theorem
In mathematical logic, the Löwenheim–Skolem theorem is a theorem on the existence and cardinality of models, named after Leopold Löwenheim and Thoralf Skolem. The precise formulation is given below. It implies that if a countable first-order theory has an infinite model, then for every infinite cardinal number ''κ'' it has a model of size ''κ'', and that no first-order theory with an infinite model can have a unique model up to isomorphism. As a consequence, first-order theories are unable to control the cardinality of their infinite models. The (downward) Löwenheim–Skolem theorem is one of the two key properties, along with the compactness theorem, that are used in Lindström's theorem to characterize first-order logic. In general, the Löwenheim–Skolem theorem does not hold in stronger logics such as second-order logic. Theorem In its general form, the Löwenheim–Skolem Theorem states that for every signature ''σ'', every infinite ''σ''-structure ''M'' and e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Countable
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time, although the counting may never finish due to an infinite number of elements. In more technical terms, assuming the axiom of countable choice, a set is ''countable'' if its cardinality (its number of elements) is not greater than that of the natural numbers. A countable set that is not finite is said countably infinite. The concept is attributed to Georg Cantor, who proved the existence of uncountable sets, that is, sets that are not countable; for example the set of the real numbers. A note on terminology Although the terms "countable" and "countably infinite" as defined here are quite comm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Bull
A bull is an intact (i.e., not castrated) adult male of the species ''Bos taurus'' (cattle). More muscular and aggressive than the females of the same species (i.e., cows), bulls have long been an important symbol in many religions, including for sacrifices. These animals play a significant role in beef ranching, dairy farming, and a variety of sporting and cultural activities, including bullfighting and bull riding. Due to their temperament, handling requires precautions. Nomenclature The female counterpart to a bull is a cow, while a male of the species that has been castrated is a ''steer'', '' ox'', or ''bullock'', although in North America, this last term refers to a young bull. Use of these terms varies considerably with area and dialect. Colloquially, people unfamiliar with cattle may refer to both castrated and intact animals as "bulls". A wild, young, unmarked bull is known as a ''micky'' in Australia.Sheena Coupe (ed.), ''Frontier Country, Vol. 1'' (Weldon R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
