Models of PA: when two elements are necessarily order automorphic

Abstract

We hope to see how much for a model M of some completion T of PA (Peano Arithmetic) does M restriction {<} determine M, say up to isomorphism. We advance in characterizing for non-standard models M of PA the "minimal" set {(a,b):n < a < b for n in N and the linear orders {c:c <_M a}, {c:c <_M b} are isomorphic}, in particular include {(a,b): for no c in M for every n in N we have M models (forall n in N)(exists c)[2 < c^n < a wedge b < a c^n]} and for some model is equal to {(a,b):a < b < a^n for some n in N}.