We are only interested in distinguishing models of PA ‘up to isomorphism’; and since any countable model is isomorphic with a model whose domain is the natural numbers, we’ll concentrate on countable models which have N as their domain. Further, we can take the interpretations of ‘O’, ‘1’ to be standard too (for any given countable model M with domain N that gives a deviant interpretation my for ‘0’ and nl for ‘1’, there will be an isomorphic model .//ll which permutes my with U and ny with 1, making compensating adjustments to the interpretations of ‘+’, ‘><’, ‘§’). So, without loss of generality, we can regard a countable model M henceforth to be a structure {N, U, 1, @9, ®, ©}.