TITLE: Constructing the natural numbers without set theory.
QUESTION [7 upvotes]: As I understand it the idea of defining everything as sets is a relatively new idea in mathematics.  Does that mean there's a non-set theoretic definition of the natural numbers?  Could there be?

REPLY [2 votes]: The natural numbers constructed in the category $\mathsf{Set}$ (objects are sets with functions as the arrows) are an example of a more general notion: that of a natural numbers object (NNO) in a topos (or a cartesian closed category). A topos can be thought of as a "universe" for doing mathematics, just as the category $\mathsf{Set}$ is used in classical foundations. The nlab article I linked to gives some explicit examples of NNO's in topoi other than $\mathsf{Set}$.