In ZF, not all "collections of objects" are sets. For example, there is no set of all sets, and there is no set of all ordinals.
So, how do we know that there is a set of all countable ordinals? In other words, how do we know that $\omega_1$ exists? (I'm assuming you don't need Choice.)
Its a consequence of the axiom of replacement. The set of all binary relations on $\omega$ exists, its just $P(\omega \times \omega)$ and the subset $X$ of all well orders of $\omega$ also exists. To every well order of $\omega$ there is a well defined ordinal number, $\omega_1$ is the image of $X$ under the assignment of ordinal numbers.
Incidentally replacement is needed even to show that $\omega+\omega$ exists.