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Full Idea
The principle of complete induction says suppose that for every number, if all the numbers less than it have a property, then so does it; it then follows that every number has the property.
Gist of Idea
Complete induction assumes for all numbers less than n, then also for n, and hence for all numbers
Source
David Bostock (Intermediate Logic [1997], 2.8)
Book Ref
Bostock,David: 'Intermediate Logic' [OUP 1997], p.48
A Reaction
Complete induction is also known as 'strong' induction. Compare Idea 13358 for 'weak' or mathematical induction. The number sequence need have no first element.
Related Idea
Idea 13358 Ordinary or mathematical induction assumes for the first, then always for the next, and hence for all [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] |