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# Topic: Zermelo set theory

###### In the News (Fri 19 Jul 19)

 Zermelo set theory - Wikipedia, the free encyclopedia Zermelo set theory, as set out in an important paper in 1908 by Ernst Zermelo, is the ancestor of modern set theory. Zermelo is of course referring to the "Russell antinomy". Zermelo's paper is notable for what may be the first mention of Cantor's theorem explicitly and by name. en.wikipedia.org /wiki/Zermelo_set_theory   (986 words)

 Class (set theory) - Wikipedia, the free encyclopedia   (Site not responding. Last check: 2007-09-17) In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. Some classes are sets, for instance the class of all integers that are even, but others are not, for instance the class of all ordinal numbers or the class of all sets. A proper class cannot be an element of a set or a class and is not subject to the Zermelo-Fraenkel axioms of set theory; thereby a number of paradoxes of naive set theory are avoided. www.bexley.us /project/wikipedia/index.php/Class_(set_theory)   (382 words)

 Zermelo-Fraenkel set theory - Wikipedia, the free encyclopedia The Zermelo-Fraenkel axioms of set theory together with the axiom of choice are the standard axioms of axiomatic set theory. Axiom of union: For any set x, there is a set y such that the elements of y are precisely the elements of the elements of x. That is, for any set x there exists a set y, such that the elements of y are precisely the subsets of x. en.wikipedia.org /wiki/Zermelo-Fraenkel_axioms   (538 words)

 Zermelo-Fraenkel set theory - Open Encyclopedia   (Site not responding. Last check: 2007-09-17) The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations. The axioms are the result of work by Thoralf Skolem in 1922, based on earlier work by Adolf Fraenkel in the same year, which was based on the axiom system put forth by Ernst Zermelo in 1908 (Zermelo set theory). Axiom of infinity: There exists a set x such that {} is in x and whenever y is in x, so is the union y ∪ {y}. open-encyclopedia.com /ZFC   (492 words)

 Set_theory   (Site not responding. Last check: 2007-09-17) Naive set theory is the original set theory developed by mathematicians at the end of the 19th century. Zermelo set theory is the theory developed by the German mathematician Ernst Zermelo. Axiomatic set theory is a rigorous axiomatic theory developed in response to the discovery of serious flaws (such as Russell's paradox) in naïve set theory. www.usedaudiparts.com /search.php?title=Set_theory   (228 words)

 Set Theory The language of set theory, in its simplicity, is sufficiently universal to formalize all mathematical concepts and thus set theory, along with Predicate Calculus, constitutes the true Foundations of Mathematics. There are four main directions of current research in set theory, all intertwined and all aiming at the ultimate goal of the theory: to describe the structure of the mathematical universe. Rather, sets are introduced either informally, and are understood as something self-evident, or, as is now standard in modern mathematics, axiomatically, and their properties are postulated by the appropriate formal axioms. plato.stanford.edu /entries/set-theory   (3302 words)

 The Factasia Glossary - Z   (Site not responding. Last check: 2007-09-17) The first axiomatisation of set theory published by Ernst Zermelo in 1908 [Zermelo08] in response to the antinomies found in informal set theory by Russell and others. Zermelo's system includes the axiom of choice, but the letter "Z" is now normally used to refer to his system with the axiom of choice omitted. Zermelo-Fraenkel set theory, an axiomatisation of set theory consisting of Zermelo set theory (see above) strengthened with the axiom of replacement, due to Abraham Fraenkel, the effect of which is to ensure that any collection of sets which can be shown to be no greater in size than an existing set is itself a set. www.rbjones.com /rbjpub/philos/glossary/z.htm   (197 words)

 2 Set Theory Axiomatic set theory has its origins in the paradoxes that plagued the naive set theory used at the turn of the century, and was first created by Zermelo in 1908 to avoid these paradoxes, but it has a greater use and appeal than its origins might suggest. Set theory has become the basis for almost all mathematics, so its axiomatization has a foundational importance. The group of axioms for both the prominent set theories is small, and mostly states the assumptions of naive set theory precisely, but there are fundamental differences, eliminating the paradoxes of the past. www.u.arizona.edu /~miller/thesis/node5.html   (445 words)

 Set Theory. Zermelo-Fraenkel Axioms. Russell's Paradox. Infinity. By K.Podnieks set theory, axioms, Zermelo, Fraenkel, Frankel, infinity, Cantor, Frege, Russell, paradox, formal, axiomatic, Russell paradox, axiom, axiomatic set theory, comprehension, axiom of infinity, ZF, ZFC The set theory adopting the axiom of extensionality (C1), the axiom C1', the separation axiom schema (C21), the pairing axiom (C22), the union axiom (C23), the power-set axiom (C24), the replacement axiom schema (C25), the axiom of infinity (C26) and the axiom of regularity (C3), is called Zermelo-Fraenkel set theory, and is denoted by ZF. The set theory ZF+AC is denoted traditionally by ZFC. www.ltn.lv /~podnieks/gt2.html   (8336 words)

 Set theory   (Site not responding. Last check: 2007-09-17) Bolzano gave examples to show that, unlike for finite sets, the elements of an infinite set could be put in 1-1 correspondence with elements of one of its proper subsets. By this stage, however, set theory was beginning to have a major impact on other areas of mathematics. Zermelo in 1908 was the first to attempt an axiomatisation of set theory. www-groups.dcs.st-and.ac.uk /~history/HistTopics/Beginnings_of_set_theory.html   (2170 words)

 Zermelo-Fraenkel Set Theory: A Supplement to Set Theory Since it is provable from this axiom and the previous axiom that there is a unique such set, we may introduce the notation ‘Ø’ to denote it. A member y of a set x with this property is called a ‘minimal’ element. In other words, if we know that φ is a functional formula (which relates each set x to a unique set y), then if we are given a set u, we can form a new set v as follows: collect all of the sets to which the members of u are uniquely related by φ. plato.stanford.edu /entries/set-theory/ZF.html   (698 words)

 The choice of a foundational system In fact the distinction between set theory and type theory is not clear-cut. From a practical point of view, one appeal of type theory is that the rendering of higher-level mathematical notions like functions is rather direct, whereas in set theory it relies on a few layers of definition. This latter theory, by the way, was the basis for an unusually formal but very elegant book by [morse-sets]; the present author been told that Morse used this formalism in his teaching and research in analysis. www.rbjones.com /rbjpub/logic/jrh0111.htm   (2043 words)

 Zermelo set theory Computer Encyclopedia Enterprise Resource Directory Complete Guide to Internet   (Site not responding. Last check: 2007-09-17) A {set theory} with the following set of {axiom}s: Extensionality: two sets are equal if and only if they have the same elements. Union: If U is a set, so is the union of all its elements. Zermelo set theory avoids {Russell's paradox} by excluding sets of elements with arbitrary properties - the Comprehension axiom only allows a property to be used to select elements of an existing set. jaysir.com /computer-encyclopedia/z/zermelo-set-theory-computer-terms.htm   (154 words)

 Zermelo-Fraenkel set theory : ZFC The Zermelo-Fraenkel axioms of set theory, denoted ZF, are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based. Axiom of extension: Two sets are the same if and only if they have the same elements. Axiom of infinity: There exists a set x such that {} is in x and whenever y is in x, so is the union y U {y}. www.fastload.org /zf/ZFC.html   (442 words)

 Zermelo-Fraenkel set theory - the free encyclopedia   (Site not responding. Last check: 2007-09-17) The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of class in addition to that of a set; it is "equivalent" in the sense that any theorem about sets which can be proved in one system can be proven in the other. www.world-knowledge-encyclopedia.com /?t=ZFC   (444 words)

 Variants of Classical Set Theory and their Applications   (Site not responding. Last check: 2007-09-17) Zermelo Fraenkel set theory with the Axiom of Choice, has been used, through much of this century, as the foundational theory for modern pure mathematics. This is axiomatic set theory, modified by dropping the Axiom of Foundation and instead adding some of a variety of possible axioms that assert the existence of non-well-founded sets. While the Axiom of Foundation is usually thought to be true for the standard iterative-combinatorial conception of set in which sets are thought of as being formed' out of their elements, the axiom plays very little role in the coding of mathematical objects. www.cs.man.ac.uk /~petera/LogicWeb/settheory.html   (509 words)

 ZF Set Theory   (Site not responding. Last check: 2007-09-17) In this set theory, no loops can occur, there is no set of all sets that do not contain themselves, and the paradox is averted. If x is a set, and there is a formula with two variables such that for every element of x, there is exactly one set which together with x makes the formula true, then there is a set of all sets which do so for some x. A theorem of Godel and Cohen proves that ZF set theory can neither prove nor disprove CH, even with the addition of the Axiom of Choice (a standard addition, giving what is called ZFC) The purpose of this page is to show exactly why and how, by the introduction of chaotic real numbers. www.wall.org /~aron/zf.html   (772 words)

 03E: Set theory Fuzzy set theory replaces the two-valued set-membership function with a real-valued function, that is, membership is treated as a probability, or as a degree of truthfulness. The theory of finite sets is, arguably, a definition of Combinatorics. Since Axiomatic Set Theory is often used to construct the natural numbers (satisfying the Peano axioms, say) it is possible to translate statements about Number Theory to Set Theory. www.math.niu.edu /~rusin/known-math/index/03EXX.html   (1585 words)

 Higher-Dimensional Algebra One of the most remarkable accomplishments of the early 20th century was to formalize all of mathematics in terms of a language with a deliberately impoverished vocabulary: the language of set theory. In Zermelo-Fraenkel set theory, everything is a set, the only fundamental relationships between sets are membership and equality, and two sets are equal if and only if they have the same elements. Thus category theory encourages a relational worldview in which things are described, not in terms of their constituents, but by their relationships to other things. math.ucr.edu /home/baez/planck/node5.html   (1538 words)

 Set Theory: Foundations of Mathematics An important part of Cantor's set theory, which forms the foundations of mathematics, is the concept of transfinite ordinals. A set theory is defined in which Generalized Continuum Hypothesis and Axiom of Choice are theorems. Axiom of Fusion is used to investigate the cardinality of the set of points in a unit interval. www.ece.rutgers.edu /~knambiar/intuitive_set_theory.html   (390 words)

 [No title] Recall that ZERMELO set theory (1908), which is essentially equivalent to the categorists' notion of ELEMENTARY TOPOS with natural numbers and the axiom of choice, is adequate for most of the purposes of mathematics, though not, as I shall try to explain, logic (and theoretical computer science). Notice that this is some two decades after the appearance of the famous "antinomies" of set theory, so presumably the set theorists' guard had dropped by that time, and they had begun again to assert megalomaniac axioms. However, in set theory due to the presence of `impredicative'' axioms the proof theoretic strength is incredibly stronger than set of Martin-Laoef type theory with \$\omega4 universes. www.mta.ca /~cat-dist/catlist/1999/zf-010499   (2571 words)

 Bibliography: Set Theory with a Universal Set This is a comprehensive bibliography on axiomatic set theories which have a universal set. Church, A. Set theory with a universal set. Sheridan, K.J. The singleton function is a set in a slight extension of Church's set theory. math.boisestate.edu /~holmes/holmes/setbiblio.html   (3632 words)

 IT@Work   (Site not responding. Last check: 2007-09-17) An important but controversial axiom which is NOT part of ZF theory is the Axiom of Choice. If letter-O has a slash across it and the zero does not, your display is tuned for a very old convention used at IBM and a few other early mainframe makers (Scandinavians curse *this* arrangement even more, because it means two of their letters collide). The register set was doubled, with two banks of registers (including A and F) that could be switched between. www.itatwork.org /jargonbuster.asp?lv=1&menuitemid=186&menuid=15&search=Z   (3327 words)

 ipedia.com: Zermelo-Fraenkel set theory Article   (Site not responding. Last check: 2007-09-17) The Zermelo-Fraenkel axioms of set theory, are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based. The Zermelo-Fraenkel axioms of set theory (ZF), are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based. An equivalent finite alternative system is given by the von Neumann-Bernays-Gödel axioms; (NBG), which distinguish between classeses and sets. www.ipedia.com /zermelo_fraenkel_set_theory.html   (516 words)

 PlanetMath: Zermelo-Fraenkel axioms Ernst Zermelo and Abraham Fraenkel proposed the following axioms as a foundation for what is now called Zermelo-Fraenkel set theory, or ZF. If this set of axioms are accepted along with the Axiom of Choice, it is often denoted ZFC. Union over a set: If is a set, then there exists a set that contains every element of each planetmath.org /encyclopedia/ZermeloFraenkelSetTheory.html   (215 words)

 Set theory   (Site not responding. Last check: 2007-09-17) is a set of axiom  s and the rules of logic for deriving theorems from those axioms. In set theory there is one primitive object, the empty set , and one relationship, set membership . The integer two is the set containing 1 and the empty set. www.mtnmath.com /book/node51.html   (192 words)

 Know it all Inc. (Z)   (Site not responding. Last check: 2007-09-17) It is based on {axiomatic set theory} and {first order predicate logic}. In other words, if you do something to each element of a set, the result is a set. Comprehension (or Restriction): If P is a {formula} with one {free variable} and X a set then {x: x is in X and P(x)} is a set. artikbre.synchro.net /docs/Z.html   (2887 words)

 A linear conservative extension of Zermelo-Fraenkel set theory - Shirahata (ResearchIndex)   (Site not responding. Last check: 2007-09-17) Abstract: In this paper, we develop the system LZF of set theory with the unrestricted comprehension in full linear logic and show that LZF is a conservative extension of ZF 0 i.e., the Zermelo-Fraenkel set theory without the axiom of regularity. However, it has been also noticed that the standard extetensionality axiom cannot be added to such a set theory without causing... A linear conservative extension of Zermelo-Fraenkel set theory. citeseer.ist.psu.edu /24614.html   (472 words)

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