Could you please help me to solve following problem from an exam.
Suppose $\langle A,\prec \rangle$ is well orderered set whose ordinal is $\alpha$.
What is the cardinal of set of all subsets of $\langle A,\prec \rangle$ that are similar (have the same ordinal) to $\langle A,\prec \rangle$.
My try so far:
If $\alpha$ is finite ordinal, than there exists only one subset of $\langle A,\prec \rangle$ that is similar to $\langle A,\prec \rangle$, $\langle A,\prec \rangle$ itself.
if $\alpha=\omega$ , then $\langle A,\prec \rangle$ is similar to $\langle N,< \rangle$, and there are at least $\aleph_0$ subsets of $\langle N,< \rangle$ that are similar to $\langle N,< \rangle$, for example, set of all natural numbers that are divided by any natural number greater than 1. Can we find more than $\aleph_0$ such sets ?
But how to proceed and form a valid argument about any ordinal $\alpha$ ?
Thank you.