Computation in Logic and Logic in Computation

TL;DR

This paper explores the decidability and definability of addition and multiplication in various number domains, extending classical theorems and establishing new results about the logical complexity of these theories.

Contribution

It extends Robinson's theorem to integers, showing addition is definable in (Z;x,<), and proves decidability of (Q;x,<) using quantifier elimination.

Findings

Addition is definable in (Z;x,<), making it undecidable.

(Q;x,<) is decidable through quantifier elimination.

Addition is not definable in (Q;x,<).

Abstract

The theory of addition in the domains of natural (N), integer (Z), rational (Q), real (R) and complex (C) numbers is decidable, so is the theory of multiplication in all those domains. By Godel's Incompleteness Theorem the theory of addition and multiplication is undecidable in the domains of N, Z and Q, though Tarski proved that this theory is decidable in the domains of R and C. The theory of multiplication and order (x,<) behaves differently in the above mentioned domains of numbers. By a theorem of Robinson, addition is definable by multiplication and order in the domain of natural numbers, thus the theory (N;x,<) is undecidable. By a classical theorem in mathematical logic, addition is not definable in terms of multiplication and order in R. In this paper, we extend Robinson's theorem to the domain of integers (Z) by showing the definability of addition in (Z;x,<), this implies that (Z;x,<) is undecidable. We also show the decidability of (Q;x,<) by the method of quantifier elimination. Whence, addition is not definable in (Q;x,<).