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Full Idea
Transfinite inductions are inductive proofs that include an extra step to show that if the statement holds for all cases less than some limit ordinal, the statement also holds for the limit ordinal.
Gist of Idea
Transfinite induction moves from all cases, up to the limit ordinal
Source
Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 5.2.1 n11)
Book Ref
Colyvan,Mark: 'An Introduction to the Philosophy of Mathematics' [CUP 2012], p.83
| 14130 | Induction is proved in Dedekind, an axiom in Peano; the latter seems simpler and clearer [Dedekind, by Russell] |
| 17855 | It may be possible to define induction in terms of the ancestral relation [Frege, by Wright,C] |
| 14125 | Finite numbers, unlike infinite numbers, obey mathematical induction [Russell] |
| 14147 | Denying mathematical induction gave us the transfinite [Russell] |
| 13358 | Ordinary or mathematical induction assumes for the first, then always for the next, and hence for all [Bostock] |
| 13359 | Complete induction assumes for all numbers less than n, then also for n, and hence for all numbers [Bostock] |
| 10603 | The logic of arithmetic must quantify over properties of numbers to handle induction [Smith,P] |
| 10891 | If a set is defined by induction, then proof by induction can be applied to it [Zalabardo] |
| 17754 | Inductive proof depends on the choice of the ordering [Walicki] |
| 17936 | Transfinite induction moves from all cases, up to the limit ordinal [Colyvan] |