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6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / f. Mathematical induction

[rule to get from axioms to general mathematical truths]

10 ideas
Finite numbers, unlike infinite numbers, obey mathematical induction [Russell]
Induction is proved in Dedekind, an axiom in Peano; the latter seems simpler and clearer [Russell]
Denying mathematical induction gave us the transfinite [Russell]
Ordinary or mathematical induction assumes for the first, then always for the next, and hence for all [Bostock]
Complete induction assumes for all numbers less than n, then also for n, and hence for all numbers [Bostock]
The logic of arithmetic must quantify over properties of numbers to handle induction [Smith,P]
It may be possible to define induction in terms of the ancestral relation [Wright,C]
If a set is defined by induction, then proof by induction can be applied to it [Zalabardo]
Inductive proof depends on the choice of the ordering [Walicki]
Transfinite induction moves from all cases, up to the limit ordinal [Colyvan]