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Full Idea
The principle of mathematical (or ordinary) induction says suppose the first number, 0, has a property; suppose that if any number has that property, then so does the next; then it follows that all numbers have the property.
Gist of Idea
Ordinary or mathematical induction assumes for the first, then always for the next, and hence for all
Source
David Bostock (Intermediate Logic [1997], 2.8)
Book Ref
Bostock,David: 'Intermediate Logic' [OUP 1997], p.48
A Reaction
Ordinary induction is also known as 'weak' induction. Compare Idea 13359 for 'strong' or complete induction. The number sequence must have a first element, so this doesn't work for the integers.
Related Idea
Idea 13359 Complete induction assumes for all numbers less than n, then also for n, and hence for all numbers [Bostock]
| 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] |