Non-derivability of Euclidean Division in $\mathrm{PA}_{\mathrm{smu}}^{-}$

Abstract

$\mathrm{PA}_{\mathrm{smu}}^{-}$ is a weak theory of arithmetic obtained by adding two principles concerning powers of two to basic axioms satisfied by the nonnegative part of a discretely ordered ring. We introduce a theory $\mathrm{PA}_{\mathrm{wit}}^{-}$ in which powers of two and the required witnesses are represented by primitive symbols. We show that, along a fixed sequence in the standard model of arithmetic, every one-variable term eventually agrees with a polynomial over the dyadic rationals. A finite avoidance lemma and the Compactness Theorem then yield a model of $\mathrm{PA}_{\mathrm{smu}}^{-}$ in which division by $3$ fails. In this model even the $n=3$ instance of the Standard Euclidean Division Principle pa16 fails. In fact, the model can be chosen to satisfy every universal $\mathcal{L}_0$-sentence true in the standard model. Consequently, neither pa16 nor the Euclidean Division Principle pa17 is derivable even after these universal truths are adjoined to $\mathrm{PA}_{\mathrm{smu}}^{-}$.