Relevant Rational Arithmetic
Abstract
The arithmetic R♯ is obtained by postulating the Peano axioms on the basis of the relevant logic R. R♯ is a remarkable arithmetic, not least in that it has finite models. In this paper we examine the options for extending R♯ from natural numbers to rational numbers, as this is the essential next step towards providing a relevant basis for mathematics and for applications. Relevant rational number theory is problematic in that the most obvious approaches lead to non-conservative extensions of R♯. We consider three ways in which relevant theories of rational arithmetic can be formulated, and note in particular how these fare in the finite models of R♯.