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Full Idea
Inductive proof is not guaranteed to work in all cases and, particularly, it depends heavily on the choice of the ordering.
Gist of Idea
Inductive proof depends on the choice of the ordering
Source
Michal Walicki (Introduction to Mathematical Logic [2012], 2.1.1)
Book Ref
Walicki,Michal: 'Introduction to Mathematical Logic' [World Scientific 2012], p.69
A Reaction
There has to be an well-founded ordering for inductive proofs to be possible.
| 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] |