Second-order arithmetic

Summary

Second-order arithmetic, often written $\mathsf{Z}_2$, is a theory with two kinds of variables. One kind ranges over natural numbers, and the other over sets of natural numbers. Its axioms are basic facts about arithmetic, induction for sets, and comprehension, which says that every definable collection of numbers is a set. A large part of ordinary mathematics, including real analysis, can be developed in it by coding real numbers and other objects as sets of numbers. Its subsystems are the setting of reverse mathematics.