TITLE: Are there infinite sets of axioms?
QUESTION [30 upvotes]: I'm reading Behnke's Fundamentals of mathematics:

If the number of axioms is finite, we can reduce the concept of a consequence to that of a tautology.

I got curious on this: Are there infinite sets of axioms? The only thing I could think about is the possible existence of unknown axioms and perhaps some belief that this number of axioms is infinite.

REPLY [2 votes]: Just to make the matter maybe a bit more clear giving another example: 
Similar to what André Nicolas stated, $ACF_0$ (the theory of algebraically closed fields of charateristic $0$) is axiomatized by $ACF$(the theory of algebraically closed fields) + $\{\neg\phi_p$ : $p$  is  prime$\}$, where $\phi_p$ says "I have characteristic $p$." $ACF$ is already not finitely axiomatizable itself but I am sure observing that $\{\neg\phi_p$ : $p$  is  prime$\}$ is an infinite set of axioms is way more obvious.