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Topic: Infinite set

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	Infinity - Wikipedia, the free encyclopedia
		An infinite set can simply be defined as one having the same size as at least one of its "proper" parts; this notion of infinity is called Dedekind infinite.
		Generalizing finite and the ordinary infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers, and transfinite sequences.
		Likewise, perpetual motion machines theoretically generate infinite energy by attaining 100% efficiency or greater, and emulate every conceivable open system; the impossible problem follows of knowing that the output is actually infinite when the source or mechanism exceeds any known and understood system.
	en.wikipedia.org /wiki/Infinity (3563 words)

	Dedekind-infinite set - Wikipedia, the free encyclopedia
		This definition of "infinite set" should be compared and contrasted to the usual definition: a set A is finite if A is empty, or if there is a positive integer n such that A is equinumerous to the set {1, 2, 3,..., n}.
		Since every infinite, well-ordered set is Dedekind-infinite, and since the AC is equivalent to the well-ordering theorem stating that every set can be well-ordered, clearly the general AC implies that every infinite set is Dedekind-infinite.
		It is notable that this definition was the first definition of "infinite" which did not rely on the definition of the natural numbers.
	en.wikipedia.org /wiki/Dedekind_infinite (821 words)

	USS Clueless - Abusing Math for God
		It's possible to define a set consisting of all the points on a line between "0" and "1" and prove that this set is infinite (aleph-one) in size, even though it's based on a finite source (a bounded line segment).
		In contemporary set theory, an actual infinite is a collection of things with an infinite number of members, for example, a library with an actually infinite set of books or a museum with an actually infinite set of paintings.
		For example, in an actually infinite set of numbers, the number of even numbers in the set is equal to all of the numbers in the set.
	denbeste.nu /cd_log_entries/2003/05/AbusingMathforGod.shtml (2219 words)

	Set Theory
		The fundamental concept in the theory of infinite sets is the cardinality of a set.
		Cantor observed that many infinite sets of numbers are countable: the set of all integers, the set of all rational numbers, and also the set of all algebraic numbers.
		The smallest infinite cardinal is the cardinality of a countable set.
	plato.stanford.edu /entries/set-theory (3302 words)

	Peter Suber, "Infinite Reflections"
		Either all infinite sets are equal in cardinality, or all infinite sets have a larger cardinality than their proper subsets, but not both.
		This innovation is due to Georg Cantor, as is set theory itself, the theory of infinite sets, and the modern concept of infinite cardinality.
		For example, an infinite set with this property will not grow in cardinality as we add members to it, one at a time (see Theorem 7 in the Appendix), and will not shrink in cardinality as we subtract members from it, one at a time (see Theorems 8 and 9 in the Appendix).
	www.earlham.edu /~peters/writing/infinity.htm (11084 words)

	The Kalam Cosmological Argument: The Question of the Metaphysical Possibility of an Infinite Set of Real Entities
		It is not at all evident that a real infinite set must be equipollent to a proper infinite subset, although both sets are equipollent to N.[37] Thus for example, let us take again the case of an infinite set of humans (each with exactly two hands) and the complementary infinite set of their hands.
		However there is a one-to-one correspondence between the sets of left hands and of right hands, and the same correspondences, respectively, between the sets of all humans with that of their right hands, or that of their left hands.
		On the other hand, the equipollence between the real infinite of all humans and that of all pairs of human hands, given that each human has exactly two hands, does not entail an equipollence between the former set and the set of all human hands.
	www.infidels.org /library/modern/arnold_guminski/kalam.shtml (8407 words)

	Roger's Web Site of Metaphysics, Physics, Cosmology and Set Theory Ideas
		Starting with a single infinite set, the goal is to determine the number of even integers relative to total number of all the positive integers within the context of this single set.
		That is, the single, original set is the "experimental" system being studied and the results obtained should be the same as when the evens are in the same, single set as the total positive integers.
		Similarities of these differing views of a single infinite set and the discrete and continuous views of the universe as seen by quantum physics and relativity, respectively, are noted.
	www.geocities.com /roger846 (2054 words)

	PlanetMath: infinite
		Assuming the Axiom of Choice (or the Axiom of Countable Choice), this definition of infinite sets is equivalent to that of Dedekind-infinite sets.
		This is version 14 of infinite, born on 2001-11-16, modified 2006-01-01.
		Object id is 881, canonical name is Infinite.
	planetmath.org /encyclopedia/InfiniteSet.html (89 words)

	Science News
		Instead, mathematicians say two infinite sets are the same size if there's a way to pair their elements, one to one, with no elements of either set left over.
		Cantor studied many infinite sets of numbers, but he never found one whose size fell between that of the counting numbers and the real numbers.
		In a book-length mathematical argument that has been percolating through the set theory community for the last few years, Woodin has proved—apart from one missing piece that must still be filled in—that elegant axioms do exist and, crucially, that every elegant axiom would make the continuum hypothesis false.
	www.phschool.com /science/science_news/articles/infinite_wisdom.html (2238 words)

	SIMMER February 1998 Presentation Topic
		For example, the set of all prime numbers, or the set of inhabitants of Ontario in January 1998.
		Two sets are the same if they have exactly the same members; for example, the set of all even prime numbers greater than 2 is the same set as the set of all unicorns, namely the empty set with no members.
		The sets of positive integers and positive even integers belong to an important class of infinite sets, the denumerable sets.
	www.math.toronto.edu /mathnet/plain/simmer/topic.feb98.html (1218 words)

	fUSION Anomaly. Infinity
		The set [1, 2, 3] cannot be matched one-to-one with any of its proper subsets; such a set is called a finite set.
		In other cases, an infinite set can be matched one-to-one only with a proper subset of another infinite set, and the second set is a "larger" infinity.
		moment exhibits infinite space, but there is a space also wherein all moments are infinitely exhibited, and the everlasting duration of infinite space is another region and room of joys.
	fusionanomaly.net /infinity.html (1773 words)

	Mathematical question of the day - bargainshare.com
		Infinite set: a set which has a proper subset that is as large as the set itself.
		Proper subset: a subset of a set that lacks at least one element that is in the set.
		It's true that both sets are infinite, but one infinite set is bigger than the other infinite set.
	www.bargainshare.com /index.php?showtopic=63112 (1486 words)

	The Kalam Cosmological Argument: The Question of the Metaphyasical Possibility of an Infinite Set of Real Entities
		The cardinality of a non-empty finite set is determined by counting, that is the process of pairing members of a finite set with the progressive sequence of natural numbers (but starting with 1) until the set is exhausted; that is, there is no remaining member to be paired with the next natural number.
		A commonly given way of illustrating the equipollences that obtain among infinite mathematical sets is to juxtapose symbols representing the set of all natural numbers with that of some other set or sets.
		It is not at all evident that a real infinite set must be equipollent to a proper infinite subset, although both sets are equipollent to N. Thus, for example, let us take again the case of an infinite set of humans (each with exactly two hands) and the complementary infinite set of their hands.
	philoonline.org /library/guminski_5_2.htm (8836 words)

	Infinite Ink: The Continuum Hypothesis by Nancy McGough
		Any infinite set of real numbers is either countably infinite or has the same cardinality as the entire set of real numbers.
		Ordering sets is important because it is the way we extend the notion of counting to infinite sets.
		Since CH is not a standard assumption in mathematics and, in fact, most set theorists think it is false, it is important for a writer to state her assumptions about CH.
	www.ii.com /math/ch (4563 words)

	Welcome to the Hotel Infinity
		At the heart of Set Theory is a hall of mirrors--the paradoxical infinity.
		Sets such as all of the counting numbers, or all of the even numbers are infinite sets.
		Although the sizes of the infinite sets of counting numbers, even numbers, odd numbers, square numbers, etc., are the same, there are other sets, the set of numbers that can be expressed as decimals, for instance, that are larger.
	www.c3.lanl.gov /mega-math/workbk/infinity/inbkgd.html (1572 words)

	iron man solo infinite set up - Shoryuken Forums (Site not responding. Last check: 2007-10-21)
		The only infinite setup off launcher I would recommend learning (besides the super jump--> infinite one I hear about) is launch, u+fp, a/d down, lp, u+fp, infinite.
		This is good because if you go in with more than one air hit on standing opponent you get infinite setup without causing flying screen like you would with the other setups mentioned.
		I pull off all the mentioned hits and all, and then I put them into the infinite, but after the initial 4 hits to setup the infinite, the count starts over when I actually start the infinite.
	www.shoryuken.com /forums/showthread.php?t=97261 (510 words)

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