Most of mathematics can be formalized in second order arithmetic, and the famous theorems proved in ACA0, which is conservative over the Peano Arithmetic.
Most relevant sets of real numbers, including all Borel sets, can be coded by real numbers with the membership relation expressible in the second order arithmetic. The primary difference between doing classical mathematics in set theory (ZFC) and doing it in second order arithmetic is that in second order arithmetic one deals with codes for sets rather than sets themselves (except sets of integers).
Under the correct formalizations, most of the general theorems are actually equivalent to the minimal canonical axiom required for their proof. Most of the basic results in analysis and algebra are provable WKL0, which has the same consistency strength as primitive recursive arithmetic and whose repertoir provably recursive functions consists of the primitive recursive functions.
Basic arithmetical theorems can be proved in exponential function arithmetic, EFA, which besides the basic axioms for addition, multiplication, and exponentiation, includes the axiom of induction for bounded quantifier formulas. EFA suffices, among other things, to prove that the theory of real closed fields, and hence of classical geometry, is complete.