It is important in the foundations of mathematics that the natural number system is characterizable as a system of 0 and a successor function by second-order logic. In other words, the following Dedekind’s second-order categoricity theorem holds: every Peano system $(P,e,F)$ is isomorphic to the natural number system $(N,0,S)$. In this talk, I will investigate Dedekind’s theorem and other similar statements. We will first do reverse mathematics over $RCA_0$, and then weaken the base theory. This is a joint work with Stephen G. Simpson.