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Topic: Universe (set theory)

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	Axiomatic set theory - Wikipedia, the free encyclopedia
		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.
		Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century.
		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.
	en.wikipedia.org /wiki/Axiomatic_set_theory (2752 words)

	Set Theory (Stanford Encyclopedia of Philosophy)
		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 (3279 words)

	Steps towards a logic of natural objects
		Set theory, among other branches of mathematical logic, has a similar ambition in that an entire universe is to be constructed -- not in this case the physical universe but the "universe of discourse" of mathematics, the aggregate of all the abstract objects studied by mathematicians.
		In set theory, it was necessitated by the early paradoxes and is articulated in a fundamental series of theorems, including those of Löwenheim-Skolem, of Gödel, and the related result of Tarski on the undefinability of truth (these are not confined to set theory).
		That each singular substance expresses the whole universe in its own way, and that in its concept are included all of the experiences belonging to it together with all of their circumstances and the entire sequence of exterior events.
	faculty.baruch.cuny.edu /lkirby/naturalobjects.html (5074 words)

	PlanetMath: universe
		In order for uncountable universes to exist, it is necessary to adopt an extra axiom for set theory.
		Finally, one must be careful when using relations within universes; the details are too technical for Bourbaki to work out (!), but see the appendix to Exposé 1 of [SGA4] for more detail.
		This is version 5 of universe, born on 2003-03-14, modified 2004-04-07.
	planetmath.org /encyclopedia/Universe.html (280 words)

	Set Theory Using the SET® Game.
		The easiest way to think of union is that for any two sets, their union includes all of the elements that are in one or both of the sets.
		The intersection of two sets is the elements that are in both sets, or the elements the two sets have in common.
		The complement of a particular set is simply all the elements in the universal set that are not in that set.
	www.setgame.com /set/set_theory.htm (836 words)

	1.1. Notation and Set Theory
		Sets are the most basic building blocks in mathematics, and it is in fact not easy to give a precise definition of the mathematical object set.
		B: A union B is the set of all elements that are either in A or in B or in both.
		B: A intersection B is the set of all elements that are in both sets A and B.
	web01.shu.edu /projects/reals/logic/notation.html (1051 words)

	New Set Theory
		Despite vast advances in set theory and mathematics in general, the language of set theory, which is also the language of mathematics, has remained the same since the beginning of modern set theory and first order logic.
		We use strong logics and weak set theories to clarify the strengths of inaccessible and Mahlo cardinals.
		It is not clear how strong the theory should be, but for the theorem the following version/axiomatization works: extensionality, foundation, empty set, pairing, union, existence of transitive closure, existence of the set of all sets with transitive closure less numerous than a given set, and bounded quantifier separation.
	web.mit.edu /dmytro/www/NewSetTheory.htm (4932 words)

	Alternative Axiomatic Set Theories (Stanford Encyclopedia of Philosophy)
		A better advertisement for the usefulness of set theory for foundations of mathematics (or at least one easier to understand for the layman) is Dedekind's definition of real numbers using "cuts" in the rational numbers (Dedekind 1872) and the definition of the natural numbers as sets due to Frege and Russell (Frege 1884).
		is the cardinality of the universe and
		Strongly cantorian sets are important because it is not necessary to assign a relative type to a variable known to be restricted to a strongly cantorian set, as it is possible to use the restriction of the singleton map and its inverse to freely adjust the type of any such variable for purposes of stratification.
	plato.stanford.edu /entries/settheory-alternative (17285 words)

	Basics of Set
		Definition (Subset): A set A is a subset of a set B if and only if everything in A is also in B.
		Definition(Empty set): A set which has no elements is called an empty set.
		Definition(Universal set): A set which has all the elements in the universe of discourse is called a universal set.
	www.cs.odu.edu /~toida/nerzic/content/set/basics.html (326 words)

	New Axioms for Set Theory
		The theory of sets is canonical and the axiom schema appears to be simply a canonical schema stating that the universe is endless and that an extension would be nonrigid, and providing powerful reflection principles for set theory.
		Finally, no definable set of real numbers negates the Continuum Hypothesis, so the negation of the CH postulates entire cardinalities of sets of real numbers, none of which are definable.
		A set of subsets of κ is a perfect subset if it is the set of paths through a perfect tree of height κ.
	web.mit.edu /dmytro/www/ProposedAxioms.htm (1541 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.
		When a universal quantification occurs as the right component of a compound formula, an additional comma is inserted in front of the space in front of the word 'for'.
	home.att.net /~zei/TMKelso/set_theory.htm (3389 words)

	SPACE.com -- Evolution: It's Only a Theory, But One Worth Teaching
		In popular culture, a "theory" is understood to be a guess or speculation that may or may not be based upon evidence and analysis.
		In science, a theory is "a well-substantiated explanation of some aspect of the natural world that can incorporate facts, laws, inferences, and tested hypotheses." (Teaching About Evolution and the Nature of Science,National Academy of Sciences, 1998: 7).
		Dismissing evolution as "only a theory" is, at the simplest level, a misunderstanding of the meaning of "theory" in science.
	www.space.com /searchforlife/seti_devore_theory_050303.html (1061 words)

	The Universe/Set Theory Conflict?
		You start with the empty set, [0] which is a subset to all sets, then you move on to the set containing the empty set, [1] and from there it goes on and on, with each additional set containing all of the previous sets.
		If numbers are sets, and the mechanisms of physics are carried out in conjunction with numbers (mathematics), then is our ideal version of the "universe" as a set containing all other sets of matter and energy, one that is not itself contained by a greater set, incorrect?
		That is nothing a priori comes with the label set - they must be shown to be sets within some model of some set theory, though which theory or model, if any, is not necessarily easy to answer.
	www.physicsforums.com /showthread.php?t=74367 (868 words)

	The Universe (Site not responding. Last check: 2007-10-10)
		If the universe is expanding, then at some time in the past, it must have started from a single point.
		Although the universe is expanding all around us, the expansion happens over such a large scale that we never notice it on Earth.
		When Einstein applied his new theory to the whole universe, he found that it predicted that space should not be stable; it should either be expanding or contracting.
	cas.sdss.org /dr3/en/proj/basic/universe (1076 words)

	QUANTUM SET THEORY INTRO
		When a singularity is predicted by or elicited from any physical theory, the logical inference to draw is that the theory fails in the locus of the singularity: singularities are not real and physical things, but rather, mathematical things.
		Quantum theory has an essential linear feature, superposition, the idea that the state of the system is generally representable 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.
	graham.main.nc.us /~bhammel/QSET/qset0.html (7445 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)

	Creating a Universe-Creation Theory
		The theory's premise involved having a universe come into being with a bug bang, expand for a while, and then implode at a certain point in time.
		The bubble universe concept involves creation of universes from the quantum foam of a "parent universe." On very small scales, the foam is frothing due to energy fluctuations.
		The universe we live in has a set of physical constants that seem tailor-made for the evolution of living things.
	web.uvic.ca /~jtwong/newtheories.htm (647 words)

	Amazon.com: Set Theory: Books: Thomas Jech (Site not responding. Last check: 2007-10-10)
		Set Theory has experienced a rapid development in recent years, with major advances in forcing, inner models, large cardinals and descriptive set theory.
		Students and researchers in set theory will find this book invaluable both as study materials and as a desktop reference.
		Intuitively, a set is a collection of all elements that satisfy a certain given property.
	www.amazon.com /Set-Theory-Thomas-Jech/dp/3540440852 (1501 words)

	Alphomism - a metaphysical belief system, without god or religion, a universe theory which generates a moral code, ...
		Alphomism - a metaphysical belief system, without god or religion, a universe theory which generates a moral code, explains the supernatural and tells the meaning of life.
		Alphomism holds that we should all be working, bonded by love and in pursuit of truth, to bring the creation of Alphoma ever closer.
		Given that each of us has our own set of priorities, and that our aims often clash with those of others, we need a set of generally agreed rules to circumvent, and sometimes deal with, conflicts of interest.
	universetheory.com /government/ruleoflaw.asp (294 words)

	set (Site not responding. Last check: 2007-10-10)
		A group of elements with three properties: (1) all elements belong to a universe, (2) either each element is a member of the set or it is not, and (3) the elements are unordered.
		Formal Definition: As an abstract data type, a set has a single query function, isIn(v, S), which tells whether an element is a member of the set or not, and two modifier functions, add(v, S) and remove(v, S).
		where S is a set and u and v are elements.
	www.nist.gov /dads/HTML/set.html (212 words)

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