Open AccessModels of Relevant Arithmetic
Author(s) -
John Slaney
Publication year - 2022
Publication title -
australasian journal of logic
Language(s) - English
Resource type - Journals
ISSN - 1448-5052
DOI - 10.26686/ajl.v19i1.6585
Subject(s) - modulo , mathematics , natural number , primitive root modulo n , arithmetic , prime (order theory) , constraint (computer aided design) , modulo operation , propositional calculus , second order arithmetic , discrete mathematics , algebra over a field , combinatorics , pure mathematics , algorithm , peano axioms , geometry
It is well known that the relevant arithmetic R# admits finite models whose domains are the integers modulo n rather than the expected natural numbers. Less well appreciated is the fact that the logic of these models is much more subtle than that of the three-valued structure in which they are usually presented. In this paper we consider the DeMorgan monoids in which R# can be modelled, deriving a fairly complete account of those modelling the stronger arithmetic RM# modulo n and a partial account for the case of R# modulo a prime number. The more general case in which the modulus is arbitrary is shown to lead to infinite propositional structures even with the additional constraint that '0=1' implies everything.