>>907
The reply >>908 wasn't me. The way I'd explain induction is the following: 

Take a lineary ordered set L. 

A subset I of L is called inductive whenever it satisfies all of the following:
a) It contains all elements less than or equal to some l in L;
b) Given an element j in I, it is greater than or equal to all elements of L or there is some element j' above j in L with [j,j'] in I;
c) If [a,b) is in I, then b is in I.
 
Induction is the theorem that if L is nonvoid and suprema of bounded above subsets of it exist, then the only inductive subset of L is L itself.

Taking L as the natural numbers, this implies that given a subset of the naturals that contains 0 and contains the successor to any of its elements, that this set contains all of the natural numbers.

Taking your subset of the naturals to be the set of all naturals with a given property, this shows how to check properties for all of the natural numbers, what is known as induction classically.

Replacing the naturals with an ordinal, you obtain what is known as transfinite induction.