I've been thinking about set theory recently, and one of the things I noticed was that if we restrict to finite sets, then the power set can be constructed through repeated application of the axioms of separation, pairing, and union. And that made me wonder how ZFC looks when we replace the Axiom of Infinity with its negation.
So I have a few different questions. For this question, "finite set theory" refers to the Axioms of Extensionality, Empty Set, Pairing, Union, Separation, and the negation of Infinity.
Can finite set theory to prove that no infinite sets exist? The negation of infinity cuts off the cumulative hierarchy at $V_\omega$, which might be enough given Foundation, but it's not entirely clear to me that the nonexistence of an inductive set implies the nonexistence of any infinite set.
Of course, the cumulative hierarchy requires the power set operation, so this raises my initial question. Is the Axiom of Power Set a theorem of finite set theory?
Are the Axioms of Replacement and Choice theorems of finite set theory? Since they both involve functions, a related question is whether functions can be constructed in finite set theory without the power set operation.
Given responses below, it seems like the Axiom of Foundation may be more important than I thought. How does it affect the answers to these if added to "finite set theory"?