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Topic: Transfinite

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	A Critique of the Diagonal Method
		This section of Descriptive Mathematics explores the paradoxes of transfinite theory and suggests that we would be better served by stopping the modern habit of treating paradoxes as a foundational issue and treat them as an interesting aside.
		Transfinite theory begins with Bolzano's interpretation of Galileo's paradox to assert that the rational numbers are denumerable, the concludes with the liars paradox to show that the reals are not.
		Transfinite theory is extremely seductive in that it gives mathematicians a feeling that they can derive the fundamental nature of the universe from contemplating paradoxes.
	descmath.com /diag (759 words)

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	NationMaster - Encyclopedia: Transfinite recursion (Site not responding. Last check: )
		Transfinite induction is the proof technique of mathematical induction when applied to (large) well-ordered sets, for instance to sets of ordinals or cardinals, or even to the class of all ordinals.
		Transfinite recursion is a notion closely related to transfinite induction, but whereas the latter is a method of proof, the former is a method of definition or construction.
		Relationship to AC There is a popular misconception that transfinite induction, or transfinite recursion, or both, require the axiom of choice.
	www.nationmaster.com /encyclopedia/Transfinite-recursion (452 words)

	transfinite number — FactMonster.com
		; the theory of transfinite numbers was introduced by Georg Cantor in 1874.
		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.
		The transfinite ordinal number of the positive integers is designated by ω.
	www.factmonster.com /ce6/sci/A0849267.html (417 words)

	PlanetMath: proof of principle of transfinite induction
		To prove the transfinite induction theorem, we note that the class of ordinals is well-ordered by
		"proof of principle of transfinite induction" is owned by jihemme.
		This is version 8 of proof of principle of transfinite induction, born on 2002-02-25, modified 2002-06-01.
	planetmath.org /encyclopedia/ProofOfPrincipleOfTransfiniteInduction.html (87 words)

	0 / 0, transfinite numbers, & related. - SciForums.com (Site not responding. Last check: )
		The subject of transfinite numbers is a bit esoteric, and I would have difficulty doing it justice in a thread at this forum.
		For example, I think that addition of transfinite numbers is viewed as the combining of two sets rather than as addition in the ordinary sense of the word.
		I was thinking of transfinite numbers as a number in laymens terms, but it is a way of measuring/numbering infinite sets by determining the size of the set.
	www.sciforums.com /showthread.php?t=51591 (1200 words)

	Transfinite number - Wikipedia, the free encyclopedia
		Transfinite numbers are cardinal numbers or ordinal numbers that are larger than all finite numbers, yet not necessarily absolutely infinite.
		The term transfinite was coined by Georg Cantor, who wished to avoid some of the implications of the word infinite in connection with these objects, which were nevertheless not finite.
		Some authors, for example Suppes, Rubin, use the term transfinite cardinal to refer to the cardinality of a Dedekind-infinite set, in contexts where this may not be equivalent to "infinite cardinal"; that is, in contexts where the axiom of countable choice is not assumed or is not known to hold.
	en.wikipedia.org /wiki/Transfinite_number (372 words)

	A Critique of the Diagonal Method
		This section of Descriptive Mathematics explores the paradoxes of transfinite theory and suggests that we would be better served by stopping the modern habit of treating paradoxes as a foundational issue and treat them as an interesting aside.
		Transfinite theory begins with Bolzano's interpretation of Galileo's paradox to assert that the rational numbers are denumerable, the concludes with the liars paradox to show that the reals are not.
		Transfinite theory is extremely seductive in that it gives mathematicians a feeling that they can derive the fundamental nature of the universe from contemplating paradoxes.
	www.descmath.com /diag (759 words)

	PlanetMath: transfinite derived series
		The transfinite derived series of a group is an extension of its derived series, defined as follows.
		The transfinite derived series eventually terminates, that is, there is some ordinal
		This is version 11 of transfinite derived series, born on 2004-03-22, modified 2006-09-15.
	www.planetmath.org /encyclopedia/Hypoabelianization.html (198 words)

	Inverse Operations With Transfinite Numbers And The Kalam Cosmological Argument
		(ii): In connection with transfinite cardinal arithmetic, we may again begin by observing that, in the Cantorian formulation of transfinite cardinal arithmetic, the inverse operations in question are indeed conventionally prohibited.
		As Craig himself notes, "addition" for transfinite cardinals is defined independently of the definition of addition for finite cardinals; consequently, it need hardly be surprising that there is no corresponding operation of "subtraction".
		In this respect, the transfinite cardinals are quite different from the transfinite ordinals: addition, multiplication, etc. are defined once and for all in Conway's system: i.e.
	www.infidels.org /library/modern/graham_oppy/t_finite.html (1349 words)

	transfinite number - Encyclopedia.com (Site not responding. Last check: )
		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.
		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.
		The transfinite ordinal number of the positive integers is designated by ω.
	www.encyclopedia.com /doc/1E1-transfin.html (499 words)

	Transfinite number Summary
		Transfinite numbers are cardinal numbers or ordinal numbers that are larger than all finite numbers, yet not necessarily absolutely infinite.
		The term transfinite was coined by Georg Cantor, who wished to avoid some of the implications of the word infinite in connection with these objects, which were nevertheless not finite.
		As with finite numbers, there are two ways of thinking of transfinite numbers, as ordinal and cardinal numbers.
	www.bookrags.com /Transfinite_number (1152 words)

	Transfinite induction - Wikipedia, the free encyclopedia
		Transfinite induction is an extension of mathematical induction to (large) well-ordered sets, for instance to sets of ordinals or cardinals.
		Transfinite recursion is a method of constructing or defining something and is closely related to the concept of transfinite induction.
		There is a popular misconception that transfinite induction, or transfinite recursion, or both, require the axiom of choice.
	en.wikipedia.org /wiki/Transfinite_induction (658 words)

	PlanetMath: transfinite induction
		The principle of transfinite induction is very similar to the principle of finite induction, except that it is stated in terms of the whole class of the ordinals.
		This is version 7 of transfinite induction, born on 2002-02-25, modified 2002-06-01.
		For an arbitrary poset, there remains only one necessary and sufficient condition to get the transfinite induction working.
	www.planetmath.org /encyclopedia/TransfiniteInduction.html (353 words)

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