Toward the Limits of the Tennenbaum Phenomenon

Abstract

We consider the theory ${\rm PA}^{＃}$ and its weak fragments in the language of arithmetic expanded with the functional symbol $＃$. We prove that ${\rm PA}^{＃}$ and its weak fragments, down to $\forall E_1^{＃}({\bf N})$ and $IE_1^{-＃}$, are subject to the Tennenbaum phenomenon with respect to $+$, $\cdot$, and $＃$. For the last two theories it is still unknown if they may have nonstandard recursive models in the usual language of arithmetic.