There are several systems of axioms we can use to base mathematics on.

A common such is ZFC, Zermaelo–Fraenkel set theory with the axiom of choice. Among its axioms are assertions like "there is a set which is empty", "there is an infinite set" and "if A and B, then in particular A".

Another is Martin-Löf type theory in one of its flavors, perhaps homotopy type theory.