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Topic: Transfinite number

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	Transfinite number - Wikipedia, the free encyclopedia
		Transfinite numbers, also known as infinite numbers, are numbers that are not finite.
		As with finite numbers, there are two ways of thinking of transfinite numbers, as ordinal and cardinal numbers.
		The continuum hypothesis states that there are no intermediate cardinal numbers between aleph-null and the cardinality of the continuum (the set of real numbers): that is to say, aleph-one is the cardinality of the set of real numbers.
	en.wikipedia.org /wiki/Transfinite_number (185 words)

	AllRefer.com - transfinite number (Mathematics) - Encyclopedia
		transfinite number, cardinal or ordinal number designating the magnitude (power) or order of an infinite set; the theory of transfinite numbers was introduced by Georg Cantor in 1874.
		, which designates the set of all subsets of the real numbers, i.e., the set of all {0,1}-valued functions whose domain is the real numbers.
		Transfinite ordinal numbers are also defined for certain ordered sets, two such being equivalent if there is a one-to-one correspondence between the sets, which preserves the ordering.
	reference.allrefer.com /encyclopedia/T/transfin.html (335 words)

	Math Forum - Ask Dr. Math
		If the set is infinite, the corresponding cardinal number is not one of the finite cardinal numbers, so it is called a transfinite (or infinite) cardinal number.
		The first transfinite number is called aleph-sub-zero (or aleph- naught, or aleph_0).
		But are there any transfinite cardinal numbers between aleph_0 and C? The name aleph_1 has been given to the smallest transfinite cardinal number larger than aleph_0.
	mathforum.org /library/drmath/view/51472.html (588 words)

	Transfinite number
		There are two kinds of transfinite numbers, ordinal and cardinal.
		The lowest transfinite number ordinal number is ω.
		The first transfinite cardinal number is aleph-null[?], the cardinality of the infinite set of the integers.
	www.fastload.org /tr/Transfinite_number.html (181 words)

	Lectures 1 & 2: An Overview of the History of Number (Site not responding. Last check: 2007-10-21)
		The problem of whether two numbers were or were not equal is not a problem in finite arithmetic; that a = a for all a is part of the definition of equality.
		We can determine by his procedures whether one transfinite number is bigger than another, but not, in effect, how much bigger, that is, whether there are other transfinite numbers between the two transfinites in question.
		Thus, when it comes to the transfinite numbers, there is a conceptual chasm between the cardinal and the ordinal-but, as we have already noted, this chasm does not exist when we consider the finite numbers.
	ruccs.rutgers.edu /mathcogn2001/numbernotes.html (2615 words)

	transfinite number -- Encyclopædia Britannica
		For example, the sets of integers, rational numbers, and real numbers are all infinite; but each is a subset of the next.
		His initial significant finding was that the set of all rational numbers is equivalent to but that the set of all real numbers is not equivalent to.
		The advance of set theory and discoveries involving infinite sets, transfinite numbers, and purely logical paradoxes caused much concern as to the foundations of mathematics.
	www.britannica.com /eb/article-9073196?tocId=9073196 (802 words)

	Math Forum - Ask Dr. Math
		I don't understand what he means by putting elements into "one-to-one" correspondences, and I also don't really understand what a "transfinite number" is. The page that I went to is: http://mathforum.org/~isaac/problems/cantor2.html If you could please explain just the very, very basics of set theory to me, I would be very grateful.
		This is sort of the situation that Cantor was in when he wanted to show that the number of integers is the same as the number of rational numbers.
		The number of rationals is the same as the number of integers.
	mathforum.org /library/drmath/view/52391.html (693 words)

	Biography of Georg Ferdinand Ludwig Philipp Cantor
		He also proved that the number of points in a line segment is equal to the number of points in an infinite line, a plane and all mathematical space.
		A transfinite number is an infinite cardinal or ordinal number.
		He did this by proving that the real numbers were not countable while he had proved that the algebraic numbers (numbers which are roots of polynomial equations with integer coefficients) were countable.
	www.andrews.edu /~calkins/math/biograph/biocanto.htm (1154 words)

	Set Theory and Logic - Numericana
		The maximum number of different results which can be obtained by changing the order of the terms in a sum of n ordinals may be tabulated as follows.
		The term surreal numbers was coined by Donald E. Knuth in 1973 to describe the beautiful construct of John H.
		The most rudimentary numbers are the counting numbers (1, 2, 3, 4...) but it's probably best to let the story begin with the natural numbers (0, 1, 2, 3...) in spite of the fact that zero is a sophisticated concept of relatively recent origin.
	home.att.net /~numericana/answer/sets.htm (3698 words)

	Transfinite number - InfoSearchPoint.com (Site not responding. Last check: 2007-10-21)
		The first transfinite cardinal number is aleph-null, $\aleph_0$ , the cardinality of the infinite set of the integers.
		The next lowest cardinal number is aleph-one, $\aleph_1$ .
		The continuum hypothesis states that there are no intermediate cardinal numbers between aleph-null and the cardinality of the real numbers (the "continuum"): that is to say, that aleph-one is the same as the cardinality of the real numbers.
	www.infosearchpoint.com /display/Transfinite_number (190 words)

	transfinite number
		; the theory of transfinite numbers was introduced by Georg Cantor in 1874.
		; e.g., the cardinal number 5 may be assigned to each of the sets {1, 2, 3, 4, 5}, {2, 4, 6, 8, 10}, {3, 4, 5, 1, 2}, and {
		The transfinite ordinal number of the positive integers is designated by ω.
	www.infoplease.com /ce6/sci/A0849267.html (367 words)

	TRANSFINITE-ABOUT US (Site not responding. Last check: 2007-10-21)
		He showed that infinite subsets of the natural numbers (such as the set of perfect squares) can be put into one- to- one correspondence with the set of natural numbers; therefore, the number of members of such subsets must be the same as the number of elements in the set of natural numbers.
		Also, with Diagonal theorem, he showed that the set of rational numbers (i.e., fractions) can be put into one-to-one correspondence with the natural numbers, and therefore has the same cardinal number as the set of natural numbers.
		The transfinite cardinal of these sets is (sometimes called E), the "smallest" transfinite number.
	www.transfinite.com /transfinite_html/ts_numbers_ht.html (190 words)

	Reviews of The New Mormon Challenge
		The difficulties that so long delayed the theory of infinite numbers were largely due to the fact that some, at least, of the inductive properties were wrongly judged to be such as must belong to all numbers; indeed it was thought that they could not be denied without contradiction.
		Nevertheless, the number of bubble universes in the multiverse may be infinite and the multiverse as a whole may be without a beginning.
		However, while the results of transfinite numbers and the concept of an infinite universe may be strange, they are not any stranger than the notions of quantum mechanics or the theory of relativity.
	www.fairlds.org /apol/TNMC/TNMC01.html (12378 words)

	Trolling in Shallow Water (Site not responding. Last check: 2007-10-21)
		Curiously, it is also the cardinality of the even natural numbers, or of the integers divisible by ten, or of the integers that are powers of two, or of the prime numbers.
		Since there is an infinite number of primes (a theorem proved by Euclid more than two thousand years ago) and since the primes are a proper subset of the set of natural numbers, it must be the case that the cardinality of the set of primes is aleph-0.
		Hence, the set of real numbers between 0 and 1 has the same cardinality as the set of all real numbers, and therefore the set of points on an infinite line has the same cardinality as the set of points on a line segment of length 1.
	shallows.blogspot.com /2004/06/musings-on-infinite.html (3030 words)

	The Kalam Cosmological Argument: The Question of the Metaphyasical Possibility of an Infinite Set of Real Entities
		(aleph zero) is also the cardinal number of all rational numbers (i.e., numbers expressible as a fraction with two integers as numerator and denominator, respectively), and of all algebraic numbers.
		The second (AV2) is that the cardinal number of N is the cardinal number of every real infinite because each such infinite is equipollent with N. The third (AV3) is that equipollence between two real infinites is a sufficient but not necessary condition for such two sets (as commonly understood) to have the same cardinality.
		The least ordinal number that can belong to a denumerable set is w, the sequent in W [the class of all ordinal numbers] of all the natural numbers, and it therefore follows.
	www.philoonline.org /library/guminski_5_2.htm (8836 words)

	Bret Willet's paper on infinity
		Cardinal numbers are those which measure the number of objects in a set, as opposed to ordinal numbers, which are numbers with a fixed predecessor and successor.
		The set of rational numbers also has a cardinality of aleph-naught, and thus is the same size as the set of integers.
		In other words, the set of all transfinite numbers is described by a cardinal number that is also transfinite [1].
	www.facstaff.bucknell.edu /udaepp/090/w3/bretw.htm (2365 words)

	The Kalam Cosmological Argument: The Question of the Metaphysical Possibility of an Infinite Set of Real Entities
		A natural number cannot be the cardinal number of the set of all natural numbers (i.e., {1, 2, 3, 4,....})[16] since there is no highest natural number.
		e.g., the cardinal number of the set of all geometrical points (on a line, in a square, or in a cube, for example), and that of the set of all geometrical curves, the latter being a cardinal number greater than the former.
		The least ordinal number that can belong to a denumerable set is ω, the sequent in W [the class of all ordinal numbers] of all the natural numbers, and it therefore follows...
	www.infidels.org /library/modern/arnold_guminski/kalam.shtml (8407 words)

	number - Wiktionary
		For his second number, he sang "The Moon Shines Bright".
		Number the baskets so that we can find them easily.
		I don't know how many books are in the library, but they must number in the thousands.
	en.wiktionary.org /wiki/number (233 words)

	natural theology > development > 2 model > 2 immensity (Site not responding. Last check: 2007-10-21)
		Even though every natural number is finite, the set of all the natural numbers is infinite, in the language of set theory 'countably infinite'.
		The permutations of the natural numbers N may be considered the elements of a set with an even greater transfinite cardinal, aleph(1).
		Second, we notice that in some respects the changing world behaves very much like a permutation process: things are swapping their positions all the time: for instance when a leaf falls the position occupied by the leaf an the tree is replaced by air, and some air near the ground is replaced by a leaf.
	www.naturaltheology.net /Development/Dev02_Model/model02Immensity.html (1209 words)

	Infinite Ink: Continuum Hypothesis Confusion
		First note that Gamow's sequence of finite and transfinite numbers is not correct.
		It has been established that the number of irrationals is the same as the number of reals, which is 2
		What we don't know about the irrationals (and the reals, which have the same cardinality as the irrationals) is where their cardinality lies in the sequence of transfinite numbers:
	ii.best.vwh.net /math/ch/confusion (331 words)

	Talk:Transfinite number - Wikipedia, the free encyclopedia
		I do think that a better method of symbolizing these transfinite numbers is needed,symboles that should calculate by virtue of their own visual characteristics.
		What sorts of things are transfinite numbers used for?
		I don't know who was the first to consider infinite numbers in Europe, but it certainly wasn't Cantor.
	en.wikipedia.org /wiki/Talk:Transfinite_number (170 words)

	QR Glossary (Site not responding. Last check: 2007-10-21)
		A rational number can be written in the form a/b, where a and b are both integers and b is not equal to zero.
		The number a (which is often one) is the lower limit of the summation, while the number b is the upper limit of the summation.
		Aleph null and aleph one are transfinite numbers.
	www.arps.org /users/hs/kochn/QR/Glossary.htm (2768 words)

	natural theology > development > 2 model > 3 a transfinite network (Site not responding. Last check: 2007-10-21)
		The number of steps required to solve some computational problems, however, grows exponentially with the size of the input.
		The maximum number is aleph(0), ie one different instruction corresponding to each different natural number.
		Although the subject of this paper is ostensibly the computable numbers, it is almost as easy to define and investigate computable functions of an integrable variable or a real or computable variable, computable predicates and so forth.
	www.naturaltheology.net /Development/Dev02_Model/model03TransNet.html (1264 words)

	The Spiritual Function of Mathematics, by Thomas J McFarlane
		In his famous diagonalization argument, Cantor proves that one transfinite number can be larger than another, in particular, that the number of points on the real number line between zero and one is greater than the number of positive integers.
		For example, construct a number that has its first digit different from the first digit of the first number, its second digit different from the second digit of the second number, its third digit different from the third digit of the third number, and so on.
		The conceptions of negative, imaginary, infinitesimal, and transfinite numbers, as well as the conception of the continuum, are all symbols for inverse cognitions by virtue of their common transcendence of the conventional universe of positive integers.
	www.integralscience.org /sacredscience/SS_spiritual.html (3319 words)

	Transfinite
		If a set A is finite, there is a nonnegative integer, denoted #A or A, which is the number of elements in A. That number is one of the finite cardinal numbers.
		Sets having this cardinal number are called countably infinite sets, or just countable sets, because they can be put into one-to-one correspondence with the positive integers, or counting numbers.
		The 1's indicate that the corresponding natural number is in the set.
	cs.wwc.edu /~aabyan/Math/Cardinality.html (708 words)

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