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Topic: Transfinite induction

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	Induction (disambiguation) - Wikipedia, the free encyclopedia
		Induction in the fields of philosophy and logic, and used in science and the scientific method.
		Mathematical induction is a method of proof in the field of mathematics.
		Transfinite induction is a kind of mathematical induction.
	en.wikipedia.org /wiki/Induction (167 words)

	Transfinite induction (Site not responding. Last check: 2007-09-17)
		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.tocatch.info /en/Transfinite_recursion.htm (612 words)

	[No title]
		This is easily proved by induction on the height of the tree.
		Note that the induction hypothesis is true for h = 2, since the polynomials of height 1 are the natural numbers, and every natural number terminates.
		Goodstein proved that tranfinite induction below the ordinal eps_0 is equivalent to convergence of all f-goodstein sequences, where an f-goodstein sequence allows a general base bumping function f:N->N. See my earlier posts for references (archived at Deja or Swarthmore).
	www.math.niu.edu /~rusin/known-math/00_incoming/goodstein (1302 words)

	Part 3: A Very Very Very Very Very Very Pathetic and Ignorant Book (Site not responding. Last check: 2007-09-17)
		The original proposals of the formalists to make classical mathematics secure by a consistency proof did not contemplate that such a method as transfinite induction up to eps0 would have to be used.
		This implies to me that all known consistency proofs of arithmetic rely on something like transfinite induction (or possibly primitive recursive functionals of finite type), the consistency of which is no more self- evident than that of arithmetic itself.
		For example, Gentzen's proof of the consistency of PA uses transfinite induction.
	www.mathpages.com /home/kmath347/kmvs03.htm (965 words)

	[No title]
		I didn't use transfinite induction, but I made heavy use of the ZFC subset axiom.
		Strong induction over the integers and transfinite induction aren't separate ZFC axioms that are independent from the others.
		Both induction results are consequences of the basic ZFC axioms....
	www.ccs.neu.edu /home/will/Ephemera/category6.txt (3378 words)

	Book's Preface (Site not responding. Last check: 2007-09-17)
		The course presented in this text concentrates on the typical methods of modern set theory: transfinite induction, Zorn's lemma, the continuum hypothesis, Martin's axiom, the diamond principle $\diamondsuit$, and elements of forcing.
		In particular, Section 6.1 illustrates a typical transfinite induction construction and the diagonalization argument by describing several constructions of the subsets of
		The overall idea behind their presentation is to introduce them as the natural refinements of the method of transfinite induction.
	www.math.wvu.edu /~kcies/Preface.html (875 words)

	2.4 Ordinal Sequences and Large Sets
		By transfinite induction, suppose the theorem holds for all
		Thus, by transfinite induction, the theorem is true.
		is clearly transitive, the general case follows by induction.
	www.u.arizona.edu /~miller/thesis/node9.html (341 words)

	Schema for transfinite induction and ordinal arithmetic (from set theory) -- Encyclopædia Britannica
		Schema for transfinite induction and ordinal arithmetic (from set theory) -- Encyclopædia Britannica
		More results on "Schema for transfinite induction and ordinal arithmetic (from set theory)" when you join.
		More from Britannica on "Schema for transfinite induction and ordinal arithmetic (from set theory)"...
	www.britannica.com /eb/article-24039?tocId=24039 (955 words)

	Rules Versus Axioms
		expresses transfinite induction along Z with respect to the formula
		comprehension rule + the hierarchy rule + the transfinite induction rule.
		9]) and so does not include the transfinite induction rule.
	www.math.psu.edu /simpson/talks/feferfest/paper2/node2.html (111 words)

	Schema for transfinite induction and ordinal arithmetic (from set theory) -- Britannica Student Encyclopedia
		Schema for transfinite induction and ordinal arithmetic (from set theory) -- Britannica Student Encyclopedia
		When Zermelo's eight were found to be inadequate for a full-blown development of transfinite induction and ordinal arithmetic, Fraenkel and Skolem independently proposed an additional axiom schema to eliminate the difficulty.
		In induction, synthesis of a specific enzyme, called an inducible enzyme (e.g.,-galactosidase in Escherichia coli), occurs when cells are exposed to the substance (substrate) upon which the enzyme acts to form a product.
	www.britannica.com /ebi/article-24039 (957 words)

	Atlas: Transfinite Induction with Control by Alan Dow (Site not responding. Last check: 2007-09-17)
		Atlas: Transfinite Induction with Control by Alan Dow
		Transfinite induction is certainly a common method in topology and, given the large number of topological examples constructed in this way, it could well be considered a "topological method".
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caad-08.
	atlas-conferences.com /c/a/a/d/08.htm (170 words)

	S98Koszmider (Site not responding. Last check: 2007-09-17)
		The method of transfinite induction is used in classical settings when one wants to build an infinite (uncountable) structure.
		But what if the method is useless due to the fact that we do not know how to handle large initial fragments of the inductive construction?
		They are sophisticated set-theoretic structures which replace the ordinals along which the usual induction is being perfored.
	www.math.wvu.edu /~kcies/Coll98S/S98Koszmider.html (156 words)

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