Are there ordinals other than the set of natural numbers which satisfy this property?

Question

Let $\alpha$ be an ordinal. We say that $\alpha$ is good iff for every $\beta\in \alpha$, there exists $\gamma\in \alpha$ such that $|\scr{P}(\beta)|\leq |\gamma|$.

Question: Is the set of natural numbers the only good ordinal ?

Thank you

Answer 1

No, the empty set also satisfies this, as do any strong limit cardinal.

Note that this condition says that $\lambda<\kappa\implies2^\lambda<\kappa$.

It is easy to show there is a proper class of strong limit cardinals, and if you are familiar with $\beth$ numbers, then the limit $\beth$'s are strong limit cardinals.