transfinite induction

plural transfinite inductions

1. mathematicsAn extension of mathematical induction to well-ordered sets of transfinite cardinality, such as sets of ordinal numbers or cardinal numbers.

   The published version of 1936 had a different proof based on the famous principle of transfinite induction up to the ordinal #92;varepsilon#95;0 by which consistency^([of Peano arithmetic]) followed. A third proof of 1938 used transfinite induction but the logical system was a sequent calculus.

   2009, Jan von Plato, “Gentzen's Logic”, in Dov M. Gabbay, John Woods, editors, Handbook of the History of Logic, Volume 5: Logic from Russell to Church, Elsevier (North-Holland), page 667

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   The validity of the principle of transfinite induction for well-ordered sets enables us to carry out proofs by transfinite induction and definitions by transfinite induction. A proof by transfinite induction is a direct application of the principle when it is required to show that each element of a well-ordered set A has a property P.[…]To understand the method of definition by transfinite induction some preparation is necessary.

   1967, Kam-Tim Leung, Doris Lai-chue Chen, Elementary Set Theory, Parts I and II, Hong Kong University Press, page 108

   1970 [Addison-Wesley], Howard DeLong, A Profile of Mathematical Logic, Dover, 2004, page 218, Just what kinds of transfinite inductions are to be considered finitary is debatable. Transfinite induction up to an arbitrary ordinal is certainly not finitary. However, it can be shown that certain transfinite inductions are reducible to ordinary mathematical inductions. For example, induction up to ω^ω is reducible to ordinary induction. Gentzen in his proof used transfinite induction up to ε₀.

   Synonyms: Noetherian induction, structural induction, well-founded induction