Does the set P(N) have a maximum?

Question

Does the power set of the natural numbers have a maximum? Basic question, but my intuition is that it does not because it is ordered by inclusion and is infinite such that the definition of maximum does not hold. Any suggestions on how to better frame this?

Answer 1

If $N$ is any set, $\mathcal P(N)$ has a maximum, which is $N$ itself, since every element of $\mathcal P(N)$ is a subset of $N$ (and also because $N\in\mathcal P(N)$).