Tennenbaum's Theorem and Unary Functions

Abstract

It is well known that in any nonstandard model of $\mathsf{PA}$ (Peano arithmetic) neither addition nor multiplication is recursive. In this paper we focus on the recursiveness of unary functions and find several pairs of unary functions which cannot be both recursive in the same nonstandard model of $\mathsf{PA}$ (e.g., $\left\{\mathrm{2x\; ,\; 2x}+1\right\}$, $\left\{x^2,2x^2\right\}$, and $\left\{2^x,3^x\right\}$). Furthermore, we prove that for any computable injection $f\left(x\right)$, there is a nonstandard model of $\mathsf{PA}$ in which $f\left(x\right)$ is recursive.