### Abstract.

We investigate the equational fragments of formal systems for arithmetic by means of the equational theory of f-rings and of their positive cones, starting from the observation that a model of arithmetic is the positive cone of a discretely ordered ring. A consequence of the discreteness of the order is the presence of a discriminator, which allows us to derive many properties of the models of our equational theories. For example, the spectral topology of discrete f-rings is a Stone topology. We also characterize the equational fragment of *Iopen*, and we obtain an equational version of Gödel's First Incompleteness Theorem. Finally, we prove that the lattice of subvarieties of the variety of discrete f-rings is uncountable, and that the lattice of filters of the countably generated distributive free lattice can be embedded into it.