How many automorphisms are there for $ \langle \omega , < \rangle $?
I'm not sure how to start this, although I expect there to be an upper bound of $2^{\aleph_0}$. ($\aleph_0^{\aleph_0} = 2^{\aleph_0}$)
Update:
Could the answer just be $1$? Since the ordinals are well ordered, it should just be rigid, right?
Suppose that $f$ is an automorphism of $\omega$ which is not the identity map. Since $\omega$ is well-ordered, there is a least $n$ such that $f(n) \neq n$. By the choice of $n$ we have $f(m) = m$ for all $m < n$. Since $f$ is one-to-one, $f(n) > n$. Since $f$ is onto, there is a $k > n$ such that $f(k) = n$. But then $n < k$ but $f(k) < f(n)$. The function $f$ is not an automorphism.