What is one set of axioms which are sufficient for Calculus? I am curious to find out what are the "minimum axioms" needed in order to be able to have highschool level math. Another way to explain it: if we were visited by a super intelligent race of aliens (assume no language barriers), which axioms would we have to give them in order for them to be able to start completely from scratch and come up with calculus as rigorously as possible? Would the axioms of logic, ZF axioms, Peano axioms be enough?
In my experience, I think you would need at least Peano's Axioms, set theory and predicate logic. Using them, I was able to formally construct the positive real numbers as Dedekind cuts.

The set theory should include rules for: subsets, power sets, Cartesian products, set equality, n-tuple equality, functions, pairwise and arbitrary union and choice. 

Your predicate logic should include rules for equality, and allow quantification over sets and functions. 

If your set theory included some kind of axiom of infinity (as does ZFC), you might be able to derive the Peano Axioms.
In my experience, I think you would need at least Peano's Axioms, set theory and predicate logic. Using them, I was able to formally construct the positive real numbers as Dedekind cuts.

The set theory should include rules for: subsets, power sets, Cartesian products, set equality, n-tuple equality, functions, pairwise and arbitrary union and choice. 

Your predicate logic should include rules for equality, and allow quantification over sets and functions. 

If your set theory included some kind of axiom of infinity (as does ZFC), you might be able to derive the Peano Axioms.