Possibilist models of executable arithmetic in an interpretation of Griss

In several recent papers, an interpretation of G.F.C. Griss’s ‘negationless’ mathematics has been given in which techniques in the style of David Nelson are used. Two refinements of this ‘executability interpretation’ have been considered: the prehensivist refinement (in which the mathematician can conceive of any statement as though true) and the nonprehensivist refinement. In this paper, we emphasize arithmetic in a prehensivist refinement. We note a connection between this type of interpretation and possibilism and use this connection to show some results concerning arithmetic, namely that mathematical induction presupposes possibilism in the sense of Luis Estrada-González and that Peano arithmetic in this system is Post consistent, i.e. has nontrivial models.