Inconsistent Models of Arithmetic Part I: Finite Models

Abstract

This paper demonstrates the existence of interpretations of the paraconsistent logic LP which model theories properly containing all the sentences of first-order arithmetic. The metatheorem about LP used, the Collapsing Lemma, takes us from any interpretation to a new one, using an equivalence relation on the domain of interpretation which is also a congruence relation on the interpretations of the function symbols of the language. This means that in a process of collapse from one interpretation to another, truth values are not lost: anything true (false) in the original interpretation is true (false) in the collapsed interpretation. One group of models use identity modulo \(n\) for some \(n\) as the consequence relation, and the successor function becomes cyclic. Another group use a relation which identifies all numbers equal to or greater than some \(n\), giving a successor relation described as a heap. The general structure of a finite model is a tail, plus a cycle of regular numbers, plus a collection of cycles of irregular numbers, such that the period of each is a divisor of the cycle of regular numbers.