This is a follow up on a comment to one of my previous questions. What's the definition of $\omega$?
Are the following equivalent definition of $\omega$:
$\omega$ is the initial ordinal of $\aleph_0$.
$\omega$ is the least/first infinite ordinal.
$\omega$ is the set of all finite ordinals.
$\omega$ is the first non-zero limit ordinal
If yes, are there any more equivalent definitions, not on this list?