So I was reading about the ZFC axioms, and apparently some of them are actually "axiom schemas." For example, there is the "axiom schema of specification," which basically says that give a set $A$ and a formula $\phi(x)$, a subset of $A$ exists where all the elements satisfy $\phi(x)$.
This is apparently not one axiom, but a schema of infinitely many axioms, because there is one axiom for every $\phi(x)$. So that must mean that for whatever reason, just letting $\phi(x)$ be an arbitrary formula does not make a valid axiom. So are there rules for what an axiom can say?
So my questions are: Why is this not allowed to be one axiom? What are the rules for what an axiom is allowed to be? And why?
from Hot Weekly Questions - Mathematics Stack Exchange
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