I'm looking at an exercise which starts as follows:
Suppose that no uncountable subset of $\mathcal P\omega$ can be well-ordered (equivalently, the Hartogs ordinal $\gamma(\mathcal P\omega)$ is $\omega_1$).
In order to proceed with the exercise I should probably understand this equivalence.
I guess that "no uncountable … can be well-ordered" means "there is one that can't be well-ordered", otherwise any finite set would be a counterexample (as they don't have such uncountable subsets yet their Hartogs ordinal is surely not $\omega_1$). Having this said, I get that the existence of an uncountable subset that can't be well-ordered implies that $\mathcal P\omega$ doesn't inject into $\omega_1$. However, why does it imply that $\omega_1$ doesn't inject into $\mathcal P\omega$?
If an ordinal can be mapped injectively into a set, then the range of such injection has an induced order isomorphic to your ordinal.
In this case we are talking about the least uncountable ordinal, or $\omega_1$.