# Uncountable Ordered Set Isomorphic to to Its Uncountable Subsets

I'm trying to show that there is an uncountable ordered set which is similar to each of its uncountable subsets.

This is what I was going for: Let $\left<A,\prec \right>$ be the set of the countable ordinals with the usual ordering on the ordinals. Any uncountable subset $B$ of $\left<A,\prec \right>$ is well ordered. One of $A$ or $B$ is similar to an initial segment of the other one. But both $A$ and $B$ have countable proper initial segments, hence the initial segment must be the whole set. In either case we get that $A$ and $B$ are similar.

I'm missing something...

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Your example is fine; it’s just your argument that needs some repair. Any proper initial segment of $A$ or $B$ must be countable (why?), so neither $A$ nor $B$ can be similar to a proper initial segment of the other, and therefore ...

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You need two facts. Any uncountable ordinal where every initial segment is countable must be $\omega_1$. And every subset of a well ordering is well ordered.

Armed with these two facts, just look at subsets, of $\omega_1$. We know they have some ordinality. What possibilities are there?

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