Date: Jan 31, 2013 12:01 AM
Author: William Elliot
Subject: Order Isomorphic

Is every infinite subset S of omega_0 with the inherited order,
order isomorphic to omega_0?

Yes. S is an ordinal, a denumerable ordinal.
Let eta be the order type of S.

Since S is a subset of omega_0, eta <= omega_0.
Since omega_0 is the smallest infinite ordinal, omega_0 <= eta.
Thus S and omega_0 are order isomorphic.

Does the same reasoning hold to show that an uncountable subset
of omega_1 with the inherited order is order isomorphic to omega_1.

It seems intuitive that since S is a subset of omega_1, that
order type S = eta <= omega_1. How could that be rigorously