Reverse Math ?
AC $\iff$ Zorn's Lemma ?
second order arithmetic
* which is actually a two-sorted first order language
* one sort of variables intended to range over natural numbers and another intended to range over sets of natural numbers
RCA_0
* RCA_0 : an ordinal base theory of reverse math.
* RCA_0 $ \supset $ first order axioms
Further reading
* Handbook of Recursive Mathematics
* Subsystems of Second Order arithmetic
* Reverse Mathematics 2001