What is the smallest infinite ordinal that is not order isomorphic to a reordering of the natural numbers?

Question

I've been working on a particular set theory problem for a while and essentially I've hit a roadblock because of this question. I just need to know what this ordinal is, because I have a sneaking suspicion that it might not be omega 1. In fact, if it did end up being a countable ordinal, it would help me immensely.

Answer 1

It is $\omega_1$. If $\omega \leq \alpha < \omega_1$, then it is countable so if $f :\omega \rightarrow \alpha$ is a bijection, then $(\omega, <_1)$ is order isomorphic to $(\alpha , \in)$ where $m \leq_1 n$ iff $f(m) \in f(n)$. What problem are you working on?

Answer 2

By the very definition, it is $\aleph_1$, which is also the smallest uncountable cardinal.

Now, you can't go further into the description of it (namely, you cannot tell whether this is or not the continuum cardinal $2^{\aleph_0}$): it has been proven that the continuum hypothesis is not decidable (that is, may be set either true of false without adding contradictions to the ZFC set of axioms)