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Full Idea
Frege's account of the ancestral has made it possible, in effect, to define the natural numbers as entities for which induction holds.
Gist of Idea
It may be possible to define induction in terms of the ancestral relation
Source
report of Gottlob Frege (Begriffsschrift [1879]) by Crispin Wright - Frege's Concept of Numbers as Objects 4.xix
Book Ref
Wright,Crispin: 'Frege's Conception of Numbers' [Scots Philosophical Monographs 1983], p.161
A Reaction
This is the opposite of the approach in the Peano Axioms, where induction is used to define the natural numbers.
| 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] |