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Ω-logic
In set theory, Ω-logic is an infinitary logic and deductive system proposed by as part of an attempt to generalize the theory of determinacy of pointclasses to cover the structure H_. Just as the axiom of projective determinacy yields a canonical theory of H_, he sought to find axioms that would give a canonical theory for the larger structure. The theory he developed involves a controversial argument that the continuum hypothesis is false. Analysis Woodin's Ω-conjecture asserts that if there is a proper class of Woodin cardinals (for technical reasons, most results in the theory are most easily stated under this assumption), then Ω-logic satisfies an analogue of the completeness theorem. From this conjecture, it can be shown that, if there is any single axiom which is comprehensive over H_ (in Ω-logic), it must imply that the continuum is not \aleph_1. Woodin also isolated a specific axiom, a variation of Martin's maximum, which states that any Ω-consistent \Pi_2 (over H_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ω-consistent Theory
In mathematical logic, an ω-consistent (or omega-consistent, also called numerically segregative) W. V. O. Quine (1971), ''Set Theory and Its Logic''. theory is a theory (collection of sentences) that is not only (syntactically) consistentS. C. Kleene, ''Introduction to Metamathematics'' (1971), p.207. Bibliotheca Mathematica: ''A Series of Monographs on Pure and Applied Mathematics'' Vol. I, Wolters-Noordhoff, North-Holland 0-7204-2103-9, Elsevier 0-444-10088-1. (that is, does not prove a contradiction), but also avoids proving certain infinite combinations of sentences that are intuitively contradictory. The name is due to Kurt Gödel, who introduced the concept in the course of proving the incompleteness theorem. Definition A theory ''T'' is said to interpret the language of arithmetic if there is a translation of formulas of arithmetic into the language of ''T'' so that ''T'' is able to prove the basic axioms of the natural numbers under this translation. A ''T'' tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinitary Logic
An infinitary logic is a logic that allows infinitely long statements and/or infinitely long proofs. The concept was introduced by Zermelo in the 1930s. Some infinitary logics may have different properties from those of standard first-order logic. In particular, infinitary logics may fail to be compact or complete. Notions of compactness and completeness that are equivalent in finitary logic sometimes are not so in infinitary logics. Therefore for infinitary logics, notions of strong compactness and strong completeness are defined. This article addresses Hilbert-type infinitary logics, as these have been extensively studied and constitute the most straightforward extensions of finitary logic. These are not, however, the only infinitary logics that have been formulated or studied. Considering whether a certain infinitary logic named Ω-logic is complete promises to throw light on the continuum hypothesis. A word on notation and the axiom of choice As a language with infin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universally Baire Set
In the mathematical field of descriptive set theory, a set of real numbers (or more generally a subset of the Baire space or Cantor space) is called universally Baire if it has a certain strong regularity property. Universally Baire sets play an important role in Ω-logic, a very strong logical system invented by W. Hugh Woodin and the centerpiece of his argument against the continuum hypothesis of Georg Cantor. Definition A subset ''A'' of the Baire space is universally Baire if it has the following equivalent properties: #For every notion of forcing, there are trees ''T'' and ''U'' such that ''A'' is the projection of the set of all branches through ''T'', and it is forced that the projections of the branches through ''T'' and the branches through ''U'' are complements of each other. #For every compact Hausdorff space Ω, and every continuous function ''f'' from Ω to the Baire space, the preimage of ''A'' under ''f'' has the property of Baire A subset A of a topological space ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuum Hypothesis
In mathematics, specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states: Or equivalently: 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 '' continuum'' for the real numbers. History Cantor believed the continuum ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Martin's Maximum
In set theory, a branch of mathematical logic, Martin's maximum, introduced by and named after Donald Martin, is a generalization of the proper forcing axiom, itself a generalization of Martin's axiom. It represents the broadest class of forcings for which a forcing axiom is consistent. Martin's maximum (\operatorname) states that if ''D'' is a collection of \aleph_1 dense subsets of a notion of forcing that preserves stationary subsets of ''ω''1, then there is a ''D''-generic filter. Forcing with a ccc notion of forcing preserves stationary subsets of ''ω''1, thus \operatorname extends \operatorname(\aleph_1). If (''P'',≤) is not a stationary set preserving notion of forcing, i.e., there is a stationary subset of ''ω''1, which becomes nonstationary when forcing with (''P'',≤), then there is a collection ''D'' of \aleph_1 dense subsets of (''P'',≤), such that there is no ''D''-generic filter. This is why \operatorname is called the maximal extension of Martin's axi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Notices Of The American Mathematical Society
''Notices of the American Mathematical Society'' is the membership journal of the American Mathematical Society (AMS), published monthly except for the combined June/July issue. The first volume was published in 1953. Each issue of the magazine since January 1995 is available in its entirety on the journal web site. Articles are peer-reviewed by an editorial board of mathematical experts. Beginning with the January 2025 issue, the editor-in-chief is Mark C. Wilson, succeeding past editor Erica Flapan. The cover regularly features mathematical visualizations. The ''Notices'' is self-described to be the world's most widely read mathematical journal. As the membership journal of the American Mathematical Society, the ''Notices'' is sent to the approximately 30,000 AMS members worldwide, one-third of whom reside outside the United States. By publishing high-level exposition, the ''Notices'' provides opportunities for mathematicians to find out what is going on in the field. Each is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Core Model
In set theory, the core model is a definable inner model of the universe of all 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 properties. Intuitively, the core model is "the largest canonical inner model there is", (here "canonical" is an undefined term)G. Sargsyan,An invitation to inner model theory. Talk slides, Young Set Theory Meeting, 2011.p. 28 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 const ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wadge Hierarchy
In descriptive set theory, within mathematics, Wadge degrees are levels of complexity for sets of reals. Sets are compared by continuous reductions. The Wadge hierarchy is the structure of Wadge degrees. These concepts are named after William W. Wadge. Wadge degrees Suppose A and B are subsets of Baire space ωω. Then A is Wadge reducible to B or A ≤W B if there is a continuous function f on ωω with A = f^ /math>. The Wadge order is the preorder or quasiorder on the subsets of Baire space. Equivalence classes of sets under this preorder are called Wadge degrees, the degree of a set A is denoted by A">math>Asub>W. The set of Wadge degrees ordered by the Wadge order is called the Wadge hierarchy. Properties of Wadge degrees include their consistency with measures of complexity stated in terms of definability. For example, if A ≤W B and B is a countable intersection of open sets, then so is A. The same works for all levels of the Borel hierarchy and the difference hier ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inaccessible Cardinal
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 c ... [...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 elementary embedding from the Von Neumann universe into a transitive inner model with critical point and . An equivalent definition is this: is Woodin |
Gödel's Completeness Theorem
Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantics, semantic truth and syntactic Provability logic, provability in first-order logic. The completeness theorem applies to any first-order Theory (mathematical logic), theory: If ''T'' is such a theory, and φ is a sentence (in the same language) and every model of ''T'' is a model of φ, then there is a (first-order) proof of φ using the statements of ''T'' as axioms. One sometimes says this as "anything true in all models is provable". (This does not contradict Gödel's incompleteness theorem, which is about a formula φu that is unprovable in a certain theory ''T'' but true in the "standard" model of the natural numbers: φu is false in some other, "non-standard" models of ''T''.) The completeness theorem makes a close link between model theory, which deals with what is true in different models, and proof theory, which studies what can be formally prov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Theory
Set theory is the branch of mathematical logic that studies Set (mathematics), 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 of set theory, 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 the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
