On the first-order theory of the remainder

TL;DR

This paper proves that the first-order theory of the structure involving natural numbers and the remainder operation is undecidable, highlighting fundamental limits of formal reasoning about division remainders.

Contribution

It establishes the undecidability of the first-order theory of (N, mod), a significant result in mathematical logic and number theory.

Findings

First-order theory of (N, mod) is undecidable.

Implications for formal reasoning about division and remainders.

Advances understanding of logical limits in number theory.

Abstract

It is proved that the first-order theory of the structure (N,mod) is undecidable. Here mod denotes the operation of computing the remainder for any division between positive integers; i.e. x mod y is the remainder obtained by the division x : y.