Existence of $\omega$ ($\aleph_0$)

Question

We have by definition $\omega = \aleph_0$ both are countable ie., successor of an integer, but $\omega$ is also limit ie., $\models (\alpha \neq 0 \wedge (\forall\beta(\beta+1 \neq \alpha)))$. Do not "countable" and "limit" contradict each other ?

Answer 1

When we say that $\omega$ is countable, what we mean is that there exists an injection $\omega\to\mathbb N$, not that it's the successor of some integer. This is certainly not the case, since as you had previously said, $\omega$ is in fact a limit ordinal. So, there's no contradiction, you just messed up the terminology somewhat.