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Set - Wikipedia, the free encyclopedia Set theory, having only been invented at the end of the 19th century, is now a ubiquitous part of mathematics education, being introduced as early as primary school. Set theory can be viewed as the foundation upon which nearly all of mathematics can be built and the source from which nearly all mathematics can be derived. denotes the set of all rational numbers (that is, the set of all proper and improper fractions). en.wikipedia.org /wiki/Set   (1394 words)

Category:Set theory - Wikipedia, the free encyclopedia Naive set theory is the original set theory developed by mathematicians at the end of the 19th century, treating sets simply as collections of things. Axiomatic set theory is a rigorous axiomatic theory developed in response to the discovery of serious flaws (such as Russell's paradox) in naive set theory. Internal set theory is an axiomatic extension of set theory that supports a logically consistent identification of illimited (enormously large) and infinitesimal elements within the real numbers. en.wikipedia.org /wiki/Category:Set_theory   (201 words)

PlanetMath: set theory Set theory is special among mathematical theories, in two ways: It plays a central role in putting mathematics on a reliable axiomatic foundation, and it provides the basic language and apparatus in which most of mathematics is expressed. A category is not a set, and a functor is not a mapping, despite similarities in both cases. This is version 8 of set theory, born on 2003-01-01, modified 2003-02-07. planetmath.org /encyclopedia/SetTheory.html   (940 words)

Naïve set theory   (Site not responding. Last check: 2007-10-12) Naïve set theory is distinguished from axiomatic set theory by the fact that the former sets as collections of objects called the elements or members of the set whereas the latter sets only as that which satisfies certain The name may be derived from the of Paul Halmos' book Naive Set Theory. Sets are of importance in mathematics ; in fact in the modern formal the whole machinery of pure mathematics (numbers relations functions etc.) is defined in terms of Naïve set theory was developed at the of the 19th century (principally by Georg Cantor and Gottlob Frege) in order to allow mathematicians to with infinite sets consistently. www.freeglossary.com /Basic_Set_Theory   (1639 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 and Logic - Numericana The collection of sets that are not members of themselves thus includes all sets and it is not a set itself. is the set of all subsets of A. A function f from set A to set B is said to be surjective when every element of B is the image of some element of A. home.att.net /~numericana/answer/sets.htm   (3734 words)

SET THEORY, QUANTUM SET THEORY & CLIFFORD ALGEBRAS The idea of quantum set theory, while it sounds to be of a mathematical nature, is necesarily of a physical nature if one means to quantize "points" that comprise sets in such a way that they are treated as physical objects with physical properties. There are two common ways of looking at classical set theory, one (symbolic logic) involves the points of some universe of discourse, and a membership relation, while the other first proceeds without a membership relation, but from a set of axioms about some pair of binary relations on a collection of sets. Formal Set theory may be mapped to a formal propositional calculus; A formal quantum set theory can be mapped to the formal propositional calculus of quantum logic as discussed long ago by Birkhoff, Jordan, von Neumann, et al., discussed in terms also of orthomodular lattices, opposed to the Boolean lattices of classical logic. graham.main.nc.us /~bhammel/QSET/qset1.html   (10711 words)

Oz's crib sheet: basic set theory   (Site not responding. Last check: 2007-10-12) Thus, 0 is the empty set Ø, n is the set {1, ..., n}, and infinity is the set {1, 2, 3, ...}. Operations on families whose index set is 2: If you think of 2 as the set {1, 2}, then 2 is just another possible index set for a family; such a family is a finite sequence of length 2, or ordered pair. Exponentiation of sets: If X and Y are sets, consider the constant family of sets whose index set is Y and whose constant value is X. math.ucr.edu /~toby/Oz/sets   (3103 words)

Amazon.ca: Axiomatic Set Theory: Books   (Site not responding. Last check: 2007-10-12) Set theory, the theory of types, and mathematical logic are still very important though in computer science and in artificial intelligence, due to the needs in these fields for knowledge representation, computational models of intelligence, and automated reasoning. The notion of a set is defined formally, and then the axiom of extensionality, which gives a criterion for two sets being equal, and the axiom schema schema of separation. The theory of denumerable sets is then discussed, followed by one of the most fascinating concepts in all of mathematics: the theory of transfinite and infinite cardinals. www.amazon.ca /exec/obidos/ASIN/0486616304   (1688 words)

Set Theory Papers of Andreas R. Blass This survey of the theory of cardinal characteristics of the continuum is to appear as a chapter in the "Handbook of Set Theory." As the title indicates, I concentrate on the combinatorial characteristics; Tomek Bartoszynski has written a chapter on the category and measure characteristics. We work in set theory without the axiom of choice, so infinite sums and products of cardinal numbers may not be well defined. In the downward-closed context, the ideal of meager sets is prime and b-complete (where b is the bounding number), while the complementary filter is g-complete (where g is the groupwise density cardinal). www.math.lsa.umich.edu /~ablass/set.html   (2137 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)

SET - The Family Game Of Visual Perception   (Site not responding. Last check: 2007-10-12) Based on set theory, this is an intriguing game of visual perception that creates excitement and draws players to the edge of their seats. It features 81 cards with one to three symbols - ovals, squiggles or diamonds - in red, green or purple that are either solid, shaded or out-lined. No luck or memory is involved and no previous knowledge is required as players race to find all the "sets" in a shifting series of 12-card layouts. www.seminarystreet.com /shopping/TT_set.htm   (86 words)

Set Theory   (Site not responding. Last check: 2007-10-12) The "cloud-capped V of infinitistic set theory" is the cumulative hierarchy of sets, starting traditionally with the empty set and built up (through systematic application of the axioms of set theory) to higher and higher orders of infinity. First, the axioms for Zermelo-Fraenkel set theory with the axiom of choice and non-set "atoms" or ur-elements are shown in full, since the version accommodating ur-elements is not found in most set theory texts. The potential advantages of NWF sets for modeling numerous aspects of reality involving "circularity" of one kind or another are extensively discussed in [Barwise and Etchemendy 87]. www.greenshade.com /sets.html   (849 words)

Metamath Solitaire   (Site not responding. Last check: 2007-10-12) This is because built into it are the axioms of logic and ZFC (Zermelo-Fraenkel with Choice) set theory, which are generally held to encompass essentially all of mathematics. Essentially everything that is possible to know in mathematics can be derived from a handful of axioms known as "Zermelo-Fraenkel set theory," which is the culmination of years of effort to isolate the essential nature of mathematics and is one of the most impressive achievements of humankind. Practically speaking, no. The axioms for set theory are presented in their raw form, and extensive logic manipulation is necessary to come up with even simple theorems of textbook set theory. us.metamath.org /mmsolitaire/mms.html   (2777 words)

Set theory and its neighbours, Thirteenth Meeting A one-day conference in the series Set theory and its neighbours but under the auspices of Cameleon, took place on Wednesday, 15th March 2006 at the Department of Mathematics, Uuniversity College London, 25 Gordon Street, London, WC1. A semifilter is a family of infinite subsets of ω, closed under taking almost supersets. Return to the Set theory and its neighbours homepage for information, including slides from the talks and related preprints, about the previous meetings. www.ucl.ac.uk /~ucahcjm/stn/stn13.html   (523 words)

Set Theory and Nominalisation, Part II - Kamareddine (ResearchIndex) Hence we develop a type theory based on Frege structures and use it as a theory of nominalisation. Kamareddine, "Set Theory and Nominalisation, Part II", Logic and Computation 2 (6) (1992) 687-707. 2 truth and set; in: J (context) - Aczel, structures et al. citeseer.ist.psu.edu /217859.html   (759 words)

The Future of Set Theory by S.Shelah Judah has asked me to speak on the future of set theory, so as the next millennium is coming, to speak on set theory in the next millennium. According to my experience, people sophisticated about mathematics with no knowledge of set theory will accept ZFC when it is presented informally (and well), including choice but not large cardinals. I will put questions on projective sets under D.2, questions on cardinal invariants for the continuum under D.3, the general partition relations of the Hungarian school and the laws of cardinal arithmetic under D.4, and the partition relations for large cardinals under D.5. shelah.logic.at /E16/E16.html   (4649 words)

Measure theory and set theory   (Site not responding. Last check: 2007-10-12) Most of this research is concerned with problems on the border of measure theory and set theory. In [P2-81] it was proved that no countable family of continuum-additive two-valued measures can cover all subsets of the reals. [KP-80] A. Krawczyk, A. Pelc, On families of sigma-complete ideals, Fundamenta Mathematicae 109 (1980), pp. w3.uqah.uquebec.ca /pelc/math.html   (454 words)

Axiomatic Set Theory A monograph containing a historical introduction by A. Fraenkel to the original Zermelo-Fraenkel form of set-theoretic axiomatics, and Paul Bernays’ independent presentation of a formal system of axiomatic set theory. By means of the Zermelo-Fraenkel system, Suppes provides best treatment of axiomatic set theory on upper undergraduate and graduate levels. Set theory and logic seen as tools for conceptual understanding of real number system. store.doverpublications.com /0486666379.html   (129 words)

Set Theory   (Site not responding. Last check: 2007-10-12) Section 4.1 from "Sets, Models, Proofs": axioms of ZFC; relations and functions; defined operations on sets. Section 4.2 from "Sets, Models, Proofs": transitive sets, ordinals, transfinite recursion on Ord, examples of ordinals, uncountable ordinals; cardinals. Rest of 8.3: stationary sets, diamond principle, construction of a Suslin line from diamond principle. www.math.uu.nl /people/jvoosten/settheory.html   (417 words)

Amazon.com: Axiomatic Set Theory: Books: Patrick Suppes   (Site not responding. Last check: 2007-10-12) Axiomatic Set Theory (AST) lays down the axioms of the now-canonical set theory due to Zermelo, Fraenkel (and Skolem), called ZFC. The reason is that subsequent presentations of set theory are too difficult, too contrived, too clever by half. In particular, he develops the theory of cardinals by means of a temporary axiom to the effect that equipollent sets have identical cardinalities. www.amazon.com /exec/obidos/tg/detail/-/0486616304?v=glance   (2791 words)

Transitive Venn diagrams with applications to the decision problem in set theory - Cantone, Omodeo, Ursino ...   (Site not responding. Last check: 2007-10-12) As a by-product, one of the core results on decidability in computable set theory is rediscovered, namely the one that regards the satis ability of unquantified set-theoretic formulae involving Boolean operators, the singleton-former, and the powerset operator. ...a well founded set theory we also have to check that the membership relation is not circular. 5 Decision procedures for elementary sublanguages of Set Theor.. citeseer.ist.psu.edu /320252.html   (579 words)

Mathematics > Mathematical Theory > Set Theory Dauben's book charts the 'road to Damascus' in Cantor's mind of the Transfinite Set Theory realm for which Cantor truly believed he was a Divinely inspired channel or 'medium' even. He came from a famous Russian musical family and was an accomplished violinist; he was an excellent illustrator in pencil. As the other reviewer has hinted, the book is set out quite differently from other mathematical texts and the reader is not presented with the dense style employed by others (eg. www.stmbook.co.uk /nBooks/570968_1.html   (930 words)

Amazon.com: Set Theory for the Working Mathematician (London Mathematical Society Student Texts): Books: Krzysztof ...   (Site not responding. Last check: 2007-10-12) Set Theory (Studies in Logic and the Foundations of Mathematics) by Kenneth Kunen This text presents methods of modern set theory as tools that can be usefully applied to other areas of mathematics. The book begins with a tour of the basics of set theory, culminating in a proof of Zorn's Lemma and a discussion of some of its applications. www.amazon.com /exec/obidos/tg/detail/-/0521594650?v=glance   (772 words)

SAGE Publications - Fuzzy Set Theory   (Site not responding. Last check: 2007-10-12) With classical sets, objects are either in the set or not, but objects can belong partially to more than one fuzzy set at a time. Many concepts in the social sciences have this characteristic, and fuzzy set theory provides methods for systematically dealing with them. A primary reason for not going beyond programmatic statements and rather unsophisticated uses of fuzzy set theory has been the lack of practical methods for combining fuzzy set concepts with statistical methods. www.sagepub.com /book.aspx?pid=11575   (163 words)

[No title]   (Site not responding. Last check: 2007-10-12) Hebrew University Combinatorics Seminar ==================== Speaker: Ehud Friedgut Title: Extremal set theory and Juntas (via discrete harmonic analysis). Date/Time: Monday, June 6th, 2005, 10:05 sharp Place: Mathematics Building, Room 209 Abstracts: Typical theorems in extremal set theory describe the structure of a family of sets, or give bounds on its size, given information on the nature of the intersections of its members. An elegant recurrent theme is showing that the extremal families have a simple structure that "essentially depends on few elements" (whatever that means.) For example the Erdos-Ko-Rado theorem states that if k www.math.technion.ac.il /~techm/20050606100520050606fri   (94 words)

Omniseek: /Science & Tech /Math /Set Theory / MATH 3523 SET THEORY Finite and infinite sets, power set, orderings, well-orderings, the axiom of choice, transfinite numbers, ordinals, axioms for set theory, models, independence results. Set theory was created almost single-handedly by Cantor in the latter part of the 19th century. Number Theory, Combinatorics, Geometry, Algebra, Calculus and Differential Equations, Probability and Statistics, Set Theory and Foundations, History of Mathematics, Physics, Music www.omniseek.com /srch/{6970}   (278 words)

SAGE Publications - Features - Fuzzy Set Theory SAGE Publications - Features - Fuzzy Set Theory Addresses issues of operationalizing graded membership in a fuzzy set and the measurement of the properties of such sets Methods for exploring the relations among sets (e.g.. www.sagepub.com /bookfeatures.aspx?pid=11575&sc=1   (47 words)

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