Models of PA: when two elements are necessarily order automorphic

TL;DR

This paper investigates how the restriction of a model of Peano Arithmetic influences its overall structure, focusing on the minimal sets that determine models up to isomorphism, especially for non-standard models.

Contribution

It characterizes the minimal sets that determine non-standard models of PA up to isomorphism, advancing understanding of model restrictions and automorphisms.

Findings

Identifies minimal sets determining models up to isomorphism.

Characterizes specific conditions for non-standard models.

Provides examples of models with particular minimal set properties.

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}.