On Sat, 15 Jul 2006 02:18:22 -0400, Hatto von Aquitanien wrote: > "With a *well* *ordered* *finite* *set* , we can count from either > end. Reversing the order makes sense, etc. We can't do that with infinite > sets."
That is nothing more than the definition of what it means for an ordinal to be infinite. One of them, anyway. In fact, it's the very one that I have been using in my discussion with Russell Easterly.
Definition. An _ordinal_ is a transitive set that is well ordered by set membership.
Definition. An ordinal is called _finite_ if it is also well ordered by the inverse of set membership. An ordinal that is not finite is called _infinite_.
Definition. A _natural number_ is a finite ordinal.
Definition. A set is called _finite_ if it can be placed in bijection with some natural number. Otherwise, the set is _infinite_.
So you have said nothing more profound than "infinite ordinals are infinite".