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Core Model
In set theory, the core model is a definable inner model of the von Neumann universe, universe of all Set (mathematics), sets. Even though set theorists refer to "the core model", it is not a uniquely identified mathematical object. Rather, it is a class of inner models that under the right set-theoretic assumptions have very special properties, most notably covering lemma, covering properties. Intuitively, the core model is "the largest canonical inner model there is" (Ernest Schimmerling and John R. Steel) and is typically associated with a large cardinal notion. If Φ is a large cardinal notion, then the phrase "core model below Φ" refers to the definable inner model that exhibits the special properties under the assumption that there does ''not'' exist a cardinal satisfying Φ. The core model program seeks to analyze large cardinal axioms by determining the core models below them. History The first core model was Kurt Gödel's constructible universe L. Ronald Jensen proved the c ... [...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] |
Woodin Cardinal
In set theory, a Woodin cardinal (named for W. Hugh Woodin) is a cardinal number \lambda such that for all functions :f : \lambda \to \lambda there exists a cardinal \kappa < \lambda with : and an : from the into a transitive with critical point and : An equivalent definition is this: ... [...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 para ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strong Limit Cardinal
In mathematics, limit cardinals are certain cardinal numbers. A cardinal number ''λ'' is a weak limit cardinal if ''λ'' is neither a successor cardinal nor zero. This means that one cannot "reach" ''λ'' from another cardinal by repeated successor operations. These cardinals are sometimes called simply "limit cardinals" when the context is clear. A cardinal ''λ'' is a strong limit cardinal if ''λ'' cannot be reached by repeated powerset operations. This means that ''λ'' is nonzero and, for all ''κ'' < ''λ'', 2''κ'' < ''λ''. Every strong limit cardinal is also a weak limit cardinal, because ''κ''+ ≤ 2''κ'' for every cardinal ''κ'', where ''κ''+ denotes the successor cardinal of ''κ''. The first infinite cardinal, (), is a strong limit cardinal, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weakly Compact Cardinal
In mathematics, a weakly compact cardinal is a certain kind of cardinal number introduced by ; weakly compact cardinals are large cardinals, meaning that their existence cannot be proven from the standard axioms of set theory. (Tarski originally called them "not strongly incompact" cardinals.) Formally, a cardinal κ is defined to be weakly compact if it is uncountable and for every function ''f'': º 2 → there is a set of cardinality κ that is homogeneous for ''f''. In this context, º 2 means the set of 2-element subsets of κ, and a subset ''S'' of κ is homogeneous for ''f'' if and only if either all of 'S''sup>2 maps to 0 or all of it maps to 1. The name "weakly compact" refers to the fact that if a cardinal is weakly compact then a certain related infinitary language satisfies a version of the compactness theorem; see below. Every weakly compact cardinal is a reflecting cardinal, and is also a limit of reflecting cardinals. This means also that weakly compact ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subcompact Cardinal
In mathematics, a subcompact cardinal is a certain kind of large cardinal number. A cardinal number ''κ'' is subcompact if and only if for every ''A'' ⊂ ''H''(''κ''+) there is a non-trivial elementary embedding j:(''H''(''μ''+), ''B'') → (''H''(''κ''+), ''A'') (where ''H''(''κ''+) is the set of all sets of cardinality hereditarily less than ''κ''+) with critical point ''μ'' and ''j''(''μ'') = ''κ''. Analogously, ''κ'' is a quasicompact cardinal if and only if for every ''A'' ⊂ ''H''(''κ''+) there is a non-trivial elementary embedding ''j'':(''H''(''κ''+), ''A'') → (''H''(''μ''+), ''B'') with critical point ''κ'' and ''j''(''κ'') = ''μ''. ''H''(''λ'') consists of all sets whose transitive closure has cardinality less than ''λ''. Every quasicompact cardinal is subcompact. Quasicompactness is a strengthening of subcompactness in that it projects large cardinal properties upwards. The relationship is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Principle
In mathematical set theory, a square principle is a combinatorial principle asserting the existence of a cohering sequence of short closed unbounded (club) sets so that no one (long) club set coheres with them all. As such they may be viewed as a kind of incompactness phenomenon. They were introduced by Ronald Jensen in his analysis of the fine structure of the constructible universe L. Definition Define Sing to be the class of all limit ordinal In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal. Alternatively, an ordinal λ is a limit ordinal if there is an ordinal less than λ, and whenever β is an ordinal less than λ, then there exists a ...s which are not regular. ''Global square'' states that there is a system (C_\beta)_ satisfying: # C_\beta is a club set of \beta. # ot(C_\beta) < \beta # If is a limit point of then and |
Stationary Subset
In mathematics, specifically set theory and model theory, a stationary set is a set that is not too small in the sense that it intersects all club sets, and is analogous to a set of non-zero measure in measure theory. There are at least three closely related notions of stationary set, depending on whether one is looking at subsets of an ordinal, or subsets of something of given cardinality, or a powerset. Classical notion If \kappa is a cardinal of uncountable cofinality, S \subseteq \kappa, and S intersects every club set in \kappa, then S is called a stationary set.Jech (2003) p.91 If a set is not stationary, then it is called a thin set. This notion should not be confused with the notion of a thin set in number theory. If S is a stationary set and C is a club set, then their intersection S \cap C is also stationary. This is because if D is any club set, then C \cap D is a club set, thus (S \cap C) \cap D = S \cap (C \cap D) is non empty. Therefore, (S \cap C) must be station ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diamond Principle
In mathematics, and particularly in axiomatic set theory, the diamond principle is a combinatorial principle introduced by Ronald Jensen in that holds in the constructible universe () and that implies the continuum hypothesis. Jensen extracted the diamond principle from his proof that the axiom of constructibility () implies the existence of a Suslin tree. Definitions The diamond principle says that there exists a , a family of sets for such that for any subset of ω1 the set of with is stationary in . There are several equivalent forms of the diamond principle. One states that there is a countable collection of subsets of for each countable ordinal such that for any subset of there is a stationary subset of such that for all in we have and . Another equivalent form states that there exist sets for such that for any subset of there is at least one infinite with . More generally, for a given cardinal number and a stationary set , the statement (some ... [...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] |
Superstrong Cardinal
In mathematics, a cardinal number κ is called superstrong if and only if there exists an elementary embedding ''j'' : ''V'' → ''M'' from ''V'' into a transitive inner model ''M'' with critical point κ and V_ ⊆ ''M''. Similarly, a cardinal κ is n-superstrong if and only if there exists an elementary embedding ''j'' : ''V'' → ''M'' from ''V'' into a transitive inner model ''M'' with critical point κ and V_ ⊆ ''M''. Akihiro Kanamori is a Japanese-born American mathematician. He specializes in set theory and is the author of the monograph on large cardinal property, large cardinals, ''The Higher Infinite''. He has written several essays on the history of mathematics, especia ... has shown that the consistency strength of an n+1-superstrong cardinal exceeds that of an n-huge cardinal for each n > 0. References * Set theory Large cardinals {{settheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strong Cardinal
In set theory, a strong cardinal is a type of large cardinal. It is a weakening of the notion of a supercompact cardinal. Formal definition If λ is any ordinal, κ is λ-strong means that κ is a cardinal number and there exists an elementary embedding ''j'' from the universe ''V'' into a transitive inner model ''M'' with critical point κ and :V_\lambda\subseteq M That is, ''M'' agrees with ''V'' on an initial segment. Then κ is strong means that it is λ-strong for all ordinals λ. Relationship with other large cardinals By definitions, strong cardinals lie below supercompact cardinals and above measurable cardinals in the consistency strength hierarchy. κ is κ-strong if and only if it is measurable. If κ is strong or λ-strong for λ ≥ κ+2, then the ultrafilter ''U'' witnessing that κ is measurable will be in ''V''κ+2 and thus in ''M''. So for any α < κ, we have ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
