Categorical characterizations of the natural numbers require primitive recursion

Abstract

Simpson and Yokoyama (2013) [9] asked whether there exists a characterization of the natural numbers by a second-order sentence which is provably categorical in the theory ${\mathsf{RCA}}_0^{⁎}$. We answer in the negative, showing that for any characterization of the natural numbers which is provably true in ${\mathsf{WKL}}_0^{⁎}$, the categoricity theorem implies ${\Sigma }_1^0$ induction.

On the other hand, we show that ${\mathsf{RCA}}_0^{⁎}$ does make it possible to characterize the natural numbers categorically by means of a set of second-order sentences. We also show that a certain ${\Pi }_2^1$-conservative extension of ${\mathsf{RCA}}_0^{⁎}$ admits a provably categorical single-sentence characterization of the naturals, but each such characterization has to be inconsistent with ${\mathsf{WKL}}_0^{⁎}+\mathrm{superexp}$.