Gene Salamin wrote:
A set is finite if there exists no bijection of it onto one of its proper subsets.
Unfortunately, that's not true. Or rather, there are models of set theory in which this definition ("Dedekind-finite") is equivalent to the usual notion of "finite", but there are others in which it is not. The Axiom of Choice is more than enough to imply they are equivalent, but that seems like much more than you want to get into if you're looking for beautiful stand-alone theorems. A set is finite if it is in bijection with one of the sets {1,...,n}. A set is infinite if it is not finite. That is, the "finite ordinals" are, by definition, the ones which you can form starting with the empty set and the construction S -> S u {S}. All other ordinals -- the ones which require the other prong of ordinal construction, the creation of limit ordinals -- are infinite. --Michael Kleber -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen.