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Topic: Positive set theory

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	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 (154 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.
		The fact that the set of all positive integers has a proper subset, like the set of squares {1, 4, 9, 16, 25,...} of the same cardinality (using modern terminology) was considered somewhat paradoxical (this had been discussed at length by Galileo among others).
	plato.stanford.edu /entries/set-theory (3302 words)

	Olivier Esser's homepage (Site not responding. Last check: 2007-11-07)
		An Interpretation of the Zermelo-Fraenkel Set Theory and the Kelley-Morse Set Theory in a Positive Theory.
		On the axiom of extensionality in the positive set theory.
		Abstract: This is the study of the relative interpretability of the axiom of extensionality in the positive set theory.
	student.vub.ac.be /~oesser/publications.html (983 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.
		Set Theory in The Mathematical Atlas by Dave Rusin (NIU).
	home.att.net /~numericana/answer/sets.htm (3709 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
		Basic set theory used to be the place where all mathematics courses started from Algebra 1 in high school to graduate level courses.
		The set A is called the domain of the function and the set B is called the codomain of the function.
		The codomain of a function f is the set of all elements that could be hit by f; the range is the set of all elements which are hit.
	www.iwu.edu /~lstout/theoremlist/node5.html (486 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.
		In dealing with a conjectured "quantum set theory", the fundamental concepts leading to classical set theory may have to be eliminated replaced or augmented.
		With Positive and negative eigenvalues one must think of a radial variable with positive and negative values so that -r is the antipodal point of +r.
	graham.main.nc.us /~bhammel/QSET/qset0.html (6822 words)

	Set Theory
		are not since the former set is a set of four objects, while the latter set is a set with only three objects, one of which itself is a set.
		Note that the empty set is a member of the universal set; it is also a subset of the universal set.
		This is read as "the set of all pairs {a, b} such that a is an element of the set A and b is an element of the set B".
	www.rwc.uc.edu /koehler/comath/26.html (1557 words)

	Set Theory (L24) (Site not responding. Last check: 2007-11-07)
		This course will cover at least what is common to them (namely historical background, ordinal arithmetic, some basic large cardinal theory and some WQO theory) but at this stage it has not been decided whether to go for the (fairly) pure set theory option (i), or the combinatorial option (ii).
		Either way, this course is a sequel to the IIB Logic, Computation and Set theory lectures of previous years.
		Drake and Singh, Intermediate Set Theory, Wiley 1996 is a paperback that may be of use.
	www.maths.cam.ac.uk /postgrad/courses/damtp/node33.html (658 words)

	Set Theory:Cardinals - Wikibooks
		When deciding how large finite sets are, we generally count the number of elements in the set, and say two sets are the same size if they have the same number of elements.
		This approach doesn't work too well if the sets are infinite, however, because we can't count the number of elements in an infinite set.
		The set of all positive integers, {1,2,3,...} is the same size as the set of nonnegative integers, {0,1,2,...}.
	en.wikibooks.org /wiki/Set_Theory:Cardinals (250 words)

	Fractal eXtreme: Fractal Theory
		For the Julia set, Z is initialized to the current point, and C is initialized to a seed value, another complex number which is typically taken from the Mandelbrot set.
		The Mandelbrot set and Julia sets are fractals.
		What this means is that the boundary between the fl area that is the Mandelbrot set and the surrounding area that isn't the Mandelbrot set is not a simple line or a curve (one dimensional), but it also isn't a filled in circle or square (two dimensional).
	www.cygnus-software.com /theory/theory.htm (1926 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		These will include von Neumann-G\"odel-Bernays (predicative) class theory, Kelley-Morse (impredicative) class theory, the set theory of Ackermann with sets and classes and powerful extensions of Jensen's NFU (New Foundations with urelements) which may not seem at first glance to be superstructures on ZFC.
		Positive set theory may also be discussed, if time permits.
		The idea of viewing set theories with universal set such as NFU or positive set theory as alternative superstructures to the usual theory of classes has been discussed earlier by Roland Hinnion.
	math.boisestate.edu /~best/best7/talks/randall.html (119 words)

	A Set Theory of Physics
		Some modern theories see the world as being in some sense a continuum and have abandoned any idea that the world is a set of atoms moving through the void.
		The primitive language of set theory, a complete description of which has just been given, is adequate for the expression of an enormous number of notions about sets and classes of sets.
		The class term 'the class of sets x such that the universe is included in x' I abbreviate as 'the empty class' (or 'f').
	home.att.net /~zei/TMKelso/set_theory.htm (3389 words)

	Axiomatic Set Theory. Zermelo-Fraenkel Axioms
		The empty set o belongs to it: ~(o in o), hence, o in R. Still, if you will consider this collection R as a set and denote it by r, then you will have the Russell's paradox: "r in r" is equivalent to ~(r in r).
		To prove the existence of uncountable sets, the power-set axiom C24 must be applied additionally: the set of all sets of natural numbers, i.e.
		The set theory adopting the axiom of extensionality (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.
	linas.org /mirrors/www.ltn.lv/2001.03.27/~podnieks/gt2.html (7448 words)

	sciforums.com - Set Theory Question
		yeah, the positive integers are a well-founded infinite poset.
		this set would not be well-founded, at least not under the normal ordering of integers.
		but in general a set may admit many orderings, and by the axiom of choice, there is always some ordering under which the set obeys the well-ordering principle, infinite or finite.
	www.sciforums.com /printthread.php?t=31450 (185 words)

	The Future of Set Theory by S.Shelah
		Position B.2 in its strong form in essence tells us that all universes are equally valid, and hence in fact we should be interested in extreme universes.
		But in the moderate sense, this position is quite complementary to the ZFC position - one approach gives the negative results for the other, so being really interested in one forces you to have some interest in the other.
		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)

	Assignments for Logic and Set Theory
		Hypothesis: In any set of n rabbits, if one of the rabbits is white, all of the rabbits are white.
		Then we are left with a set of n rabbits, one of which is white.
		Since we know that all of the rabbits in this set, with the possible exception of Bunny, are white, we have a set of n rabbits with one of the rabbits being white.
	matcmadison.edu /alehnen/logicasn/prologic.html (2569 words)

	Set theory and physics
		Four speculations about the relevance of set theoretical modeling for physics are presented: the rôle of transcendental set theory (i) in chaos theory, (ii) for paradoxical decompositions of solid threedimensional objects, (iii) in the theory of effective computability (Church-Turing thesis) related to the possible ``solution of supertasks'', and (iv) for weak solutions.
		Several approaches to set theory and their advantages and disadvantages for physical applications are discussed: Cantorian ``naive'' (i.e., non-axiomatic) set theory, constructivism and operationalism.
		Kreisel, A notion of mechanistic theory, Synthese 29(1974), 11-26.
	tph.tuwien.ac.at /~svozil/publ/set.htm (6923 words)

	Notes on Set Theory and Logic
		Let A be the set of all positive integers less than 20.
		Let B be the set of all positive integers that end in 7 or 2.
		Let C be the set of all multiples of 3.
	www.math.umn.edu /~jhall/courses/2474/hw1.htm (187 words)

	Problems: Set Theory and Notation
		Prove or disprove: if the square of a number is an odd integer, then the original number must also be an odd integer.
		Let A be the set N x N and define a relation r on N x N by saying that (a, b) is related to (a’, b’) if a * b’ = a’ * b.
		Give examples to indicate that the resulting set of all equivalence classes has all of the familiar properties of the rational numbers (and hence can serve to define the rationals based only on the natural numbers).
	pirate.shu.edu /projects/reals/logic/problems.html (921 words)

	Toronto Set Theory Seminar (Site not responding. Last check: 2007-11-07)
		For a Baire topological space X, the algebra of sets having the property of Baire modulo the ideal of first category sets is analogous, from some points of view, to the measure algebra of a measure space.
		In particular, the theory of liftings for such algebras has many points in common with the theory of liftings for measure spaces, including many of the same open problems.
		The definition of this class of groups is obtained by suitably localizing to the identity element of the notion of amenability.
	www.math.yorku.ca /~ifarah/seminar.html (1048 words)

	set theory book errata (Site not responding. Last check: 2007-11-07)
		Delete the sentence "Since t is a positive rational, e is also a positive integer."
		Add a part (d): Show that C-bar is the smallest transitive set that includes C.
		That is, whenever D is a transitive set including C, then C-bar is a subset of D.
	www.math.ucla.edu /~hbe/est/errata.html (187 words)

	Set Theory (M24) (Site not responding. Last check: 2007-11-07)
		This course is a sequel to the IIB Logic Computation and Set theory lectures of previous years.
		The course will differ slightly from that of the same title given in 2001/2 because of the lack this year of courses in Model theory and Lambda calculus.
		Some of the more recondite parts of Set Theory will be edged out in favour of more mainstream topics in Logic--which nevertheless retain intimate connections to set theory.
	www.maths.cam.ac.uk /CASM/courses/02-03/descriptions/node27.html (418 words)

	Set Point Theory (Site not responding. Last check: 2007-11-07)
		For those of you who always seem to be dieting and not getting desired results, I have some good news and some bad news.
		The bad news is that your body may just not be designed to be as lean as you wish it to be.
		The set point theory states that we are each born with a tendency toward a certain individualized body weight range.
	bionicgroup.com /entertainment/cafe/thedeanteam/setpoint.html (168 words)

	Set Theory Logic and Foundations Math
		A Crash Course in the Mathematics of Infinite Sets
		Site desc: A weak version of ordinary set theory using bounded quantification.
		This is a refinement of Russell's theory of types based on the observation that the types in Russell's theory look the same, as far as one can apparently prove.
	www.xmeta.com /web/26922/science/math/logic-and-foundations/set-theory (985 words)

	Positive Sayings Set 5
		We act as though comfort and luxury were the chief requirements of life, when all that we need to make us really happy is something to be enthusiastic about.
		When enthusiasm is inspired by reason, controlled by caution, sound in theory, practical in application, reflects confidence, spreads good cheer, raises morale, inspires associates, arouses loyalty, and laughs at adversity, it is beyond price.
		They say a person needs just three things to be truly happy in this world.
	www.angelfire.com /ca4/angelwing1/positive5.html (566 words)

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