<logic> A powerful technique for proving that a theorem holds
for all cases in a large or infinite well-ordered sets. The
proof has two steps, the basis and induction step. Roughly, in
the basis the theorem is proved to hold for the "ancestor" case,
and in the induction step it is proved to hold for all
"descendant" cases. Example: 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.) For a more precise definition, see sub-items below. See
heredity, induction Proof that