The Theory of Ceers Computes True Arithmetic

Abstract

We show that the theory of the partial order of computably enumerable equivalence relations (ceers) under computable reduction is 1-equivalent to true arithmetic. We show the same result for the structure comprised of the dark ceers and the structure comprised of the light ceers. We also show the same for the structure of $\mathcal{I}$-degrees in the dark, light, or complete structure. In each case, we show that there is an interpretable copy of $(\mathbb{N},+,\cdot )$.