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Order Isomorphic
Posted:
Jan 31, 2013 12:01 AM


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 shown?



