induction
A method of proving statements about {well-ordered
sets}. If S is a well-ordered set with ordering "<", and we
want to show that a property P holds for every element of S,
it is sufficient to show that, for all s in S,
IF for all t in S, t < s ◦> P(t) THEN P(s)
I.e. if P holds for anything less than s then it holds for s.
In this case we say P is proved by induction.
The most common instance of proof by induction is induction
over the {natural numbers} where we prove that some property
holds for n◦0 and that if it holds for n, it holds for n:1.
(In fact it is sufficient for "<" to be a {well-founded}
{partial order} on S, not necessarily a well-ordering of S.)
(1999-12-09)