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Topic: Alternative set theory

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	Axiomatic set theory - Wikipedia, the free encyclopedia
		Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century.
		Initially controversial, set theory has come to play the role of a foundational theory in modern mathematics, in the sense of a theory invoked to justify assumptions made in mathematics concerning the existence of mathematical objects (such as numbers or functions) and their properties.
		The most frequent objection to set theory is the constructivist view that mathematics is loosely related to computation and that naive set theory is being formalised with the addition of noncomputational elements.
	www.wikipedia.org /wiki/Axiomatic_set_theory (2654 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 (175 words)

	Alternative set theory - Wikipedia, the free encyclopedia
		Generically, an alternative set theory is an alternative mathematical approach to the concept of set.
		It is a proposed alternative to standard set theory.
		Specifically, Alternative Set Theory (or AST) refers to a particular set theory developed in the 1970s and 1980s by Petr Vopěnka and his students.
	en.wikipedia.org /wiki/Alternative_set_theory (135 words)

	Positive set theory - Wikipedia, the free encyclopedia
		The axiom of closure: for every set x, a set exists which is the intersection of all sets containing x; this is called the closure of x and is written {x}.
		The universal set is a proper set in this theory.
		The theory can interpret ZFC (by restricting oneself to the set of sets whose complement is also a set).
	en.wikipedia.org /wiki/Positive_set_theory (199 words)

	[No title]
		There is a set I and relation P such that "for each x in I, there is exactly one y in I such that Pxy" and "for each y in I, there is at most one x such that Pxy" and "for some x, for all y, not Pyx".
		The infinite sets are exactly the classes such that there are class bijections witnessing the fact that they are the same size as I. Theorem: The class of finite ordinals is a set.
		The countable ordinals are all sets; the proper class ordinal is aleph_one.
	math.boisestate.edu /~holmes/holmes/pocket.txt (910 words)

	PlanetMath: concepts in set theory
		The aim of this entry is to present a list of the key objects and concepts used in set theory.
		The reader should take care that if the objects under discussion are not just sets (say, groups or schemes) the operations may not be simple set operations, but rather their analogue in the relevant category.
		This is version 44 of concepts in set theory, born on 2004-02-29, modified 2005-04-14.
	planetmath.org /encyclopedia/NotationInSetTheory.html (357 words)

	PlanetMath: von Neumann-Bernays-Gödel set theory
		This theory is essentially stronger than ZFC or NBG, as it can prove their consistency (in addition to everything they already prove).
		by means of the Burali-Forti paradox, that the class of all ordinals is not a set, and hence there is a bijection between the class of ordinals and the class of all sets.
		This is version 12 of von Neumann-Bernays-Gödel set theory, born on 2003-06-25, modified 2004-09-22.
	planetmath.org /encyclopedia/VonNeumannBernausGodelSetTheory.html (795 words)

	Set Theory and Logic - Numericana (Site not responding. Last check: 2007-10-20)
		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.
		Set Theory in The Mathematical Atlas by Dave Rusin (NIU).
	home.att.net /~numericana/answer/sets.htm (3709 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 (10320 words)

	Structuralism, Category Theory and Philosophy of Mathematics
		In set theory these are also known as order-preserving bijections, and in category theory functors are generalizations of isomorphisms that allow us to "translate" from one category to another in a way that preserves the categorial structure of its source.
		This is due to the fact that set theory is extensional, and the combinatorial aspects of mathematics, which is concerned with the finitely presented properties of the inscriptions of the formal language is intensional.
		Set theory strips away structure from the ontology of mathematics leaving pluralities of structureless individuals open to the imposition of new structure.
	www.mmsysgrp.com /strctcat.htm (7237 words)

	LC '98 abstract: Pavol Zlato\v s (Site not responding. Last check: 2007-10-20)
		AST is, first of all, an alternative theory of the {\it infinity}, which is treated as a phenomenon accompanying our views towards the {\it horizon\/}.
		From the formal point of view, AST is closely related to nonstandard analysis\,---\,it can be regarded as the theory of a nostandard saturated universe of power $\aleph_1$ of hereditarily finite sets with two sorts of variables\,---\,for sets (elements of the universe) and classes (parts of the universe).
		In such a space, every infinite set contained in the {\it galaxy\/} of $0$, such that no couple of its elements are infinitesimally close, contains an infinite subset of indiscernibles, such that no of \,its elements is infinitesimally close to the subspace spanned by the remaining ones.
	www.math.cas.cz /~lc98/abstracts/Zlatos.html (388 words)

	Annotated Bibliography: Theory of Semisets
		The theory of semiset was proposed and developed by Vopenka and Hajek to represent sets with imprecise boundaries.
		It is a complicated theory and are more general than the fuzzy set theory.
		The relationship between semisets and fuzzy sets was studied by Novak.
	ils.unc.edu /~wongs/info_retrieval/annotated/semiset.html (56 words)

	Category Theory
		For instance, given two sets A and B, set theory allows us to construct their cartesian product A X B. For an example of the second sort, given a finite abelian group, it can be decomposed into a product of some of its subgroups.
		For it is in his thesis that Lawvere proposed the idea of developing the category of categories as a foundation for category theory, set theory and, thus, the whole of mathematics, as well as using categories for the study of theories, that is the logical aspects of mathematics.
		Given these simple facts, it remains to be seen whether category theory should be "on the same plane", so to speak, with set theory, whether it should be considered seriously as providing a foundational alternative to set theory or whether it is foundational in a different sense altogether.
	plato.stanford.edu /entries/category-theory (7029 words)

	QUANTUM SET THEORY INTRO
		Quantum theory has an essential feature, superposition, the idea that the state of the system is generally represented by a complex linear combination of mutually exclusive alternatives.
		The cardinality of the set of mutually exclusive alternatives, becomes the dimension of a complex linear space whose elements represent exhaustively the states of the system.
		In dealing with a conjectured "quantum set theory", the fundamental concepts leading to classical set theory may have to be eliminated replaced or augmented.
	graham.main.nc.us /~bhammel/QSET/qset0.html (6822 words)

	Axiom schema of specification : Axiom of separation (Site not responding. Last check: 2007-10-20)
		The axiom schema of specification is generally considered uncontroversial as far as it goes, and it or an equivalent appears in just about any alternative axiomatization of set theory.
		Then the set B guaranteed by the axiom of replacement is precisely the set B required for the axiom of specification.
		But in this case, the set B required for the axiom of specification is the empty set, so the axiom schema follows in general using also the axiom of empty set.
	www.eurofreehost.com /ax/Axiom_of_separation_2.html (393 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 (8335 words)

	Descriptive Set Theory
		Group theory serves as a common logic for theories investigating mathematical structures that are subtypes of groups.
		In the descriptive set theory the meaning of "simple", "definable" sets (of real numbers) is defined explicitly by introducing the so-called Borel sets and projective sets.
		The class of projective sets is closed is closed under finite unions, finite intersections, and inverse continuous images, yet (unlike the class of Borel sets) it is not closed under countable unions and countable intersections.
	linas.org /mirrors/www.ltn.lv/2001.03.27/~podnieks/gtaa.html (3789 words)

	Bibliography: Set Theory with a Universal Set
		This is a comprehensive bibliography on axiomatic set theories which have a universal set.
		Set theory and hierarchy theory, Springer Lecture Notes in Mathematics 619, pp.
		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)

	LC '98 abstract: P.V. Andreyev, E.I. Gordon (Nizhny Novgorod, Russia) (Site not responding. Last check: 2007-10-20)
		The main difference between that version and the theory presented here is that the last can be viewed as a theory of formula classes of Bounded Set Theory (BST) by V. Kanovei [2] rather than that of Internal Set Theory (IST) by E. Nelson, in particular every set is an element of a standard set.
		A class is called p-standard (or standard relative to the set p) iff it can be represented as a cut of a standard class by the set p.
		First of all there are semisets (proper subclasses of sets) here and a theorem can be proved that a class is standard iff its intersection with any standard set is a standard set.
	www.math.cas.cz /~lc98/abstracts/Andreyev.html (407 words)

	Douglas L. Grant - Alternative universes: the role of set theory in topological algebra (Site not responding. Last check: 2007-10-20)
		Douglas L. Grant - Alternative universes: the role of set theory in topological algebra
		Alternative universes: the role of set theory in topological algebra
		This survey will explore some of the theorems of topological algebra in which significantly more powerful results are available than in the analagous case for Hausdorff spaces.
	www.cms.math.ca /Events/winter98/w98-abs/node168.f?nomenu=1 (109 words)

	Axiom of pairing (Site not responding. Last check: 2007-10-20)
		The axiom of pairing is generally considered uncontroversial, and it or an equivalent appears in just about any alternative axiomatization of set theory.
		This set C is again unique by the axiom of extension, and is denoted {A
		Of course, we can't refer to a finite number of sets rigorously without already having in our hands a (finite) set to which the sets in question belong.
	www.eurofreehost.com /ax/Axiom_of_pairing_2.html (296 words)

	Cogprints - HYPERSOLVER: A Graphical Tool for Commonsense Set Theory (Site not responding. Last check: 2007-10-20)
		This paper investigates an alternative set theory (due to Peter Aczel) called Hyperset Theory.
		Aczel uses a graphical representation for sets and thereby allows the representation of non-well-founded sets.
		This may be a useful tool for commonsense reasoning.
	cogprints.org /466 (120 words)

	Encyclopedia: Positive set theory
		People who viewed "Positive set theory" also viewed:
		Updated 201 days 12 hours 30 minutes ago.
		In mathematical logic, positive set theory is an alternative set theory consisting of the following axioms:
	www.nationmaster.com /encyclopedia/Positive-set-theory (206 words)

	The Uncertain Reasoning Research Group -Manchester (Site not responding. Last check: 2007-10-20)
		Aspects of uncertainty studied by the Alternative Set Theory.
		Any sufficiently robust belief set or knowledge base will almost certainly be inconsistent.
		If we are to make progress towards the formalization of practical reasoning, we must learn to deal with inconsistent information in a non-trivial way.
	www.ma.man.ac.uk /logic/ur/people.html (226 words)

	Descriptive Set Theory. By K.Podnieks (Site not responding. Last check: 2007-10-20)
		you need not to treat one-dimensional, two-dimensional, three-dimensional sets etc. separately.
		To build sets in ZF you can use only comprehension axioms (see Section 2.3).
		We cannot hope to extend this theorem to PI sets:
	www.ltn.lv /~podnieks/gtaa.html (1770 words)

	OUP: Set Theory and its Philosophy: Potter (Site not responding. Last check: 2007-10-20)
		Comprehensive treatment of the basics of set theory
		More in the same subject area: Mathematics; Philosophy of mathematics; Mathematical foundations; Mathematical logic; Set theory;
		The specification in this catalogue, including without limitation price, format, extent, number of illustrations, and month of publication, was as accurate as possible at the time the catalogue was compiled.
	www4.oup.co.uk /isbn/0-19-926973-4 (257 words)

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