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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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