I have the following list of sufficient conditions for showing that a set or class $M$ models each of the axioms of first-order ZFC. I would like to know how they would change if we wanted a model of second-order ZFC. I'm guessing that since only the axioms of Separation and Replacement (and possibly Foundation) would change, all the other conditions would remain sufficient?
If it's not possible to give similar sufficient conditions for proving the second order versions of those axioms, how does one approach a proof that a set or class models them?
Best Answer
These axioms actually already say that $M$ is a model of second-order ZFC. Statements (iii) and (v) are the second-order versions of Separation and Replacement (the first-order version of (iii) would only say that subsets of $x$ which are definable in $M$ must be elements of $M$, for instance).
(Note that depending on how you define "second-order powerset", (viii) might be weaker, since you might demand that the actual powers set is an element of $M$. However, if you are additionally assuming (iii), it makes no difference, since $P(x)\cap M=M$ in that case.)