transfinite induction
The proof schema corresponding to ordinary mathematical induction taken into the transfinite, i.e. defined over orderings that may be larger than the standard ordering of the set of all natural numbers. Gentzen showed that if this operation is allowed (involving a relaxation of the finitary methods of theories subject to Gödel's incompleteness theorem for arithmetic), we may prove the consistency and completeness of arithmetic. But the Gödel results then apply to the mathematics of the transfinite. See also finitism.

Philosophy dictionary. . 2011.

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