Minimum number of axioms for ZFC set theory What is the minimum number of axioms needed for ZFC Set Theory?
I have found that Suppes' Axiomatic Set Theory lists 7 axioms, but I am
not sure if this can be reduced or not.  Any references on this subject would be much appreciated.
Suppes' list:


*

*Axiom of Extensionality

*Sum Axiom

*Power Set Axiom

*Axiom of Regularity

*Axiom of Infinity

*Axiom Schema of Replacement

*Axiom of Choice

 A: It seems to me the OP is asking whether any of the 7 assertions (specifically 6 axioms and a schema) listed are unnecessary, i.e. whether we can remove any one of them and still derive all of ZFC (similar to how Suppes removed pairing and comprehension because they can be proved from the other axioms). And the answer to that is no, none of these 7 assertions can be proven from the other 6. E.g., ZFC-Infinity+ $\neg$Infinity holds in the model $HF,$ and is thus relatively consistent to ZFC.
A: The reflection principle is a theorem schema in ZFC, meaning that for each formula $\phi(\vec x)$ we can prove in ZFC a version of the principle for $\phi$. In particular, it gives us that if $\phi$ holds (in the universe of sets) then there is some ordinal $\alpha$ such that $V_\alpha\models \phi$. 
It follows from this that (assuming its consistency) $\mathsf{ZFC}$ is not finitely axiomatizable. Otherwise, $\mathsf{ZFC}$ would prove its own consistency, violating the second incompleteness theorem. The  (standard) list of axioms you presented is actually an infinite list, with replacement being in fact an axiom schema (one axiom for each formula). 

It is perhaps worth mentioning that no appeal to the incompleteness theorem is needed: If $\mathsf{ZFC}$ is consistent, and finitely axiomatizable, then it would prove (because of reflection) that there are $\alpha$ such that $V_\alpha\models\mathsf{ZFC}$. It would then follow that there is a least such $\alpha$. But inside $V_\alpha$ there must be some $\beta$ such that $$V_\alpha\models\mbox{``}V_\beta\models\mathsf{ZFC}\mbox{''}$$
(because $V_\alpha$ is a model of set theory, so it satisfies reflection), and easy absoluteess arguments give us that then $\beta<\alpha$ is indeed an ordinal, and $V_\beta$ is really a model of $\mathsf{ZFC}$, contradicting the minimality of $\alpha$. 
