The first limit-ordinal $\omega $ is the least-upper bound of the finite ordinals. There is no greatest finite-ordinal. So $\omega $ is not among the finite ordinals. But why then suppose that the finite ordinals have a least-upper bound at all?
Suppose it is because the finite ordinals are collected as a set. Then this set itself witnesses the existence of $\omega$. Doesn't this beg the question? In virtue of what are we entitled to assume that the finite ordinals are collectable?