|
Positive Set Theory
In mathematical logic, positive set theory is the name for a class of alternative set theories in which the axiom of comprehension holds for at least the positive formulas \phi (the smallest class of formulas containing atomic membership and equality formulas and closed under conjunction, disjunction, existential and universal quantification). Typically, the motivation for these theories is topological: the sets are the classes which are closed under a certain topology. The closure conditions for the various constructions allowed in building positive formulas are readily motivated (and one can further justify the use of universal quantifiers bounded in sets to get generalized positive comprehension): the justification of the existential quantifier seems to require that the topology be compact. Axioms The set theory \mathrm^+_\infty of Olivier Esser consists of the following axioms: Extensionality \forall x \forall y (\forall z (z \in x \leftrightarrow z \in y) \to x = y) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and analysis. In the early 20th century it was shaped by David Hilbert's program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in proving consistency. Work in set theory sho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Von Neumann
John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest coverage of any mathematician of his time and was said to have been "the last representative of the great mathematicians who were equally at home in both pure and applied mathematics". He integrated pure and applied sciences. Von Neumann made major contributions to many fields, including mathematics (foundations of mathematics, measure theory, functional analysis, ergodic theory, group theory, lattice theory, representation theory, operator algebras, matrix theory, geometry, and numerical analysis), physics (quantum mechanics, hydrodynamics, ballistics, nuclear physics and quantum statistical mechanics), economics ( game theory and general equilibrium theory), computing ( Von Neumann architecture, linear programming, numerical meteo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Olivier Esser '', a 1992 drama film
{{disambiguation ...
Olivier is the French form of the given name Oliver. It may refer to: * Olivier (given name), a list of people and fictional characters * Olivier (surname), a list of people * Château Olivier, a Bordeaux winery *Olivier, Louisiana, a rural populated place in the United States * Olivier (crater), on the Moon * Olivier salad, a popular dish of Russian cuisine * ''Olivier'' (novel), the first published novel by French author Claire de Duras * The Olivier Theatre (named after the actor Laurence Olivier), one of three auditoria at the Royal National Theatre * The Laurence Olivier Awards, a theatrical award * Olivier (comics), a foe of The Punisher See also * ''Olivier, Olivier ''Olivier, Olivier'' is a 1992 drama film directed by Agnieszka Holland. It entered the competition at the 49th Venice International Film Festival and won an award at the 1992 Valladolid International Film Festival. The plot involves a nine-year-o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alonzo Church
Alonzo Church (June 14, 1903 – August 11, 1995) was an American mathematician, computer scientist, logician, philosopher, professor and editor who made major contributions to mathematical logic and the foundations of theoretical computer science. He is best known for the lambda calculus, the Church–Turing thesis, proving the unsolvability of the Entscheidungsproblem, the Frege–Church ontology, and the Church–Rosser theorem. He also worked on philosophy of language (see e.g. Church 1970). Alongside his student Alan Turing, Church is considered one of the founders of computer science. Life Alonzo Church was born on June 14, 1903, in Washington, D.C., where his father, Samuel Robbins Church, was a Justice of the Peace and the judge of the Municipal Court for the District of Columbia. He was the grandson of Alonzo Webster Church (1829-1909), United States Senate Librarian from 1881-1901, and great grandson of Alonzo Church, a Professor of Mathematics and Astronomy and 6t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isaac Malitz
Isaac Richard Jay Malitz (born 1947, in Cleveland, Ohio) is a logician who introduced the subject of positive set theory in his 1976 Ph.D. Thesis at UCLA The University of California, Los Angeles (UCLA) is a public land-grant research university in Los Angeles, California. UCLA's academic roots were established in 1881 as a teachers college then known as the southern branch of the California .... References Isaac (Richard) Jay Malitz– entry in the Mathematics Genealogy Project 1947 births Living people American logicians University of California, Los Angeles alumni Scientists from Cleveland Date of birth missing (living people) 20th-century American mathematicians {{US-mathematician-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Set
In set theory, a universal set is a set which contains all objects, including itself. In set theory as usually formulated, it can be proven in multiple ways that a universal set does not exist. However, some non-standard variants of set theory include a universal set. Reasons for nonexistence Many set theories do not allow for the existence of a universal set. There are several different arguments for its non-existence, based on different choices of axioms for set theory. Regularity In Zermelo–Fraenkel set theory, the axiom of regularity and axiom of pairing prevent any set from containing itself. For any set A, the set \ (constructed using pairing) necessarily contains an element disjoint from \, by regularity. Because its only element is A, it must be the case that A is disjoint from \, and therefore that A does not contain itself. Because a universal set would necessarily contain itself, it cannot exist under these axioms. Russell's paradox Russell's paradox prevents the ex ... [...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 ZFC, 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'': [κ] 2 → there is a Set (mathematics), set of cardinality κ that is Homogeneous (large cardinal property), homogeneous for ''f''. In this context, [κ] 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'']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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morse–Kelley Set Theory
In the foundations of mathematics, Morse–Kelley set theory (MK), Kelley–Morse set theory (KM), Morse–Tarski set theory (MT), Quine–Morse set theory (QM) or the system of Quine and Morse is a first-order axiomatic set theory that is closely related to von Neumann–Bernays–Gödel set theory (NBG). While von Neumann–Bernays–Gödel set theory restricts the bound variables in the schematic formula appearing in the axiom schema of Class Comprehension to range over sets alone, Morse–Kelley set theory allows these bound variables to range over proper classes as well as sets, as first suggested by Quine in 1940 for his system ML. Morse–Kelley set theory is named after mathematicians John L. Kelley and Anthony Morse and was first set out by and later in an appendix to Kelley's textbook ''General Topology'' (1955), a graduate level introduction to topology. Kelley said the system in his book was a variant of the systems due to Thoralf Skolem and Morse. Morse's ow ... [...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 isomorph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom Of Infinity
In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory. It guarantees the existence of at least one infinite set, namely a set containing the natural numbers. It was first published by Ernst Zermelo as part of his set theory in 1908.Zermelo: ''Untersuchungen über die Grundlagen der Mengenlehre'', 1907, in: Mathematische Annalen 65 (1908), 261-281; Axiom des Unendlichen p. 266f. Formal statement In the formal language of the Zermelo–Fraenkel axioms, the axiom reads: :\exists \mathbf \, ( \empty \in \mathbf \, \land \, \forall x \in \mathbf \, ( \, ( x \cup \ ) \in \mathbf ) ) . In words, there is a set I (the set which is postulated to be infinite), such that the empty set is in I, and such that whenever any ''x'' is a member of I, the set formed by taking the union of ''x'' with its singleton is also a member of I. Such a set is sometimes called an inductive set. I ... [...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] |
Von Neumann–Bernays–Gödel Set Theory
In the foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of Zermelo–Fraenkel–choice set theory (ZFC). NBG introduces the notion of class, which is a collection of sets defined by a formula whose quantifiers range only over sets. NBG can define classes that are larger than sets, such as the class of all sets and the class of all ordinals. Morse–Kelley set theory (MK) allows classes to be defined by formulas whose quantifiers range over classes. NBG is finitely axiomatizable, while ZFC and MK are not. A key theorem of NBG is the class existence theorem, which states that for every formula whose quantifiers range only over sets, there is a class consisting of the sets satisfying the formula. This class is built by mirroring the step-by-step construction of the formula with classes. Since all set-theoretic formulas are constructed from two kinds of atomic formulas ( membership and equali ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
