transfinite induction


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transfinite induction

[tranz′fī‚nīt in′dək·shən]
(mathematics)
A reasoning process by which if a theorem holds true for the first element of a well-ordered set N and is true for an element n whenever it holds for all predecessors of n, then the theorem is true for all members of N.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.

transfinite induction

(mathematics)
Induction over some (typically large) ordinal.
This article is provided by FOLDOC - Free Online Dictionary of Computing (foldoc.org)
References in periodicals archive ?
We shall build / using transfinite induction. For every [alpha] < c, we shall define [x.sub.[alpha]] [member of] B and f([x.sub.[alpha]]), in such way that (([x.sub.[alpha]]/([x.sub.[alpha]])) [member of] [K.sup.[alpha]].
to the schema of transfinite induction in [L.sub.PA] up to [alpha] adjoined to PA.
The fact that Gentzen's proof can be interpreted in the natural numbers shows that the arithmetical statement which corresponds via coding to the theorem validating transfinite induction as far as [[Epsilon].sub.0] is not provable in PA, since, if it were, PA would indeed prove its own consistency, contrary to Godel's second incompleteness theorem.