It is known that the Axioms of ZFC are not necessarily independent of each other. For example, it can be shown that one can derive Separation from Replacement even though both are listed as axioms of ZFC. My question is simply this:
Can it be shown that Powerset is not derivable from the other axioms of ZFC?
Yes. Perhaps the most natural examples of models of ZFC - Powerset are the sets hereditarily of cardinality less than a regular cardinal. More precisely, we define:
$H(\kappa) = \{x: |tc(x)|<\kappa\}$
When $\kappa$ is regular, it is easy to check that $H(\kappa) \vDash ZFC - Powerset$. But, when $\kappa$ is a successor cardinal, it is also easy to see that $H(\kappa) \vDash \neg Powerset$.
No, it is not. In particular, notice that none of the other axioms will ever allow you to construct a set with cardinality greater than $\mathbb N$. For most of the axioms, this is fairly obvious. The only particularly dangerous one is the axiom of union; however, this axiom cannot necessarily create larger sets than it is given - in particular, if we only have countable sets, then we can only create the countable union of countable sets through axiom of union (since the set over which we take the union must be countable, as must each element thereof), which, assuming AC, is countable. Therefore, the axiom of powerset, which implies that there are sets larger than $\mathbb N$ must not follow from the other axioms.
