Ω-logic

	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) consistent (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'' that interprets arithmetic is ω-inconsistent if, for some property ''P'' of natural numbers (defined by a formula in the language of ''T''), ''T'' proves ''P''(0), ''P''(1), ''P''(2), and so on (that is, for every standard natural number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Infinitary Logic An infinitary logic is a logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ... that allows infinitely long statement (logic), statements and/or infinitely long Mathematical proof, proofs. Some infinitary logics may have different properties from those of standard first-order logic. In particular, infinitary logics may fail to be Compactness (logic), compact or Completeness (logic), 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 system, Hilbert-type infinitary logics, as these have been extensively studied and constitute the most straightforward extensions of finitary logi ... [...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 spa ... [...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]
	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 (MM) 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 MM 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 MM is called the maximal extension of Martin's axiom. The existence of a supercompact ... [...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 appeared 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. Since 2019, the editor-in-chief is Erica Flapan. The cover regularly features mathematical visualization Mathematical phenomena can be understood and explored via visualization. Classically this consisted of two-dimensional drawings or building three-dimensional models (particularly plaster models in the 19th and early 20th century), while today it ...s. 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	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]
	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.html" ;"title="math>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 th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Inaccessible Cardinal In set theory, an uncountable cardinal is inaccessible if it cannot be obtained from smaller cardinals by the usual operations of cardinal arithmetic. More precisely, a cardinal is strongly inaccessible if it is uncountable, it is not a sum of fewer than cardinals smaller than , and \alpha < \kappa implies $2^ < \kappa$ . The term "inaccessible cardinal" is ambiguous. Until about 1950, it meant "weakly inaccessible cardinal", but since then it usually means "strongly inaccessible cardinal". An uncountable cardinal is weakly inaccessible if it is a regular weak limit cardinal. It is strongly inaccessible, or just inaccessible, if it is a regular strong limit cardinal (this is equivalent to the definition given above). Some authors do not require weakly and strongly ... [...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 : $\ \subseteq \kappa$ and an elementary embedding : $j : V \to M$ from the Von Neumann universe $V$ into a transitive inner model $M$ with critical point $\kappa$ and : $V_ \subseteq M.$ An equivalent definition is this: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Gödel's Completeness Theorem Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic. The completeness theorem applies to any first-order 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 universally true is provable". This does not contradict Gödel's incompleteness theorem, which shows that some formula φu is unprovable although true in the natural numbers, which are a particular model of a first-order theory describing them — φu is just false in some other model of the first-order theory being considered (such as a non-standard model of arithmetic for Peano arithmetic). It makes a close link between model theory that deals with what is true in different models, and proof theory tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	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]