On highly equivalent non-isomorphic countable models of arithmetic and set theory

TL;DR

This paper investigates the degree of similarity between non-isomorphic countable models of arithmetic and set theory by examining the length of back-and-forth sequences, revealing models with arbitrarily large Scott heights.

Contribution

It demonstrates that for any countable ordinal, there exist non-isomorphic models of PA and ZFC with back-and-forth sequences of that length, extending understanding of model similarity.

Findings

Existence of models with arbitrarily large back-and-forth sequence lengths.

Models of PA and ZFC can have Scott heights exceeding any given countable ordinal.

The measure of model closeness can be characterized by back-and-forth sequence length.

Abstract

It is well-known that the first order Peano axioms PA have a continuum of non-isomorphic countable models. The question, how close to being isomorphic such countable models can be, seems to be less investigated. A measure of closeness to isomorphism of countable models is the length of back-and-forth sequences that can be established between them. We show that for every countable ordinal alpha there are countable non-isomorphic models of PA with a back-and-forth sequence of length alpha between them. This implies that the Scott height (or rank) of such models is bigger than $\alpha$. We also prove the same result for models of ZFC.