Submission date: 27 August 2022

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.

Mathematics Subject Classification: 03C62, 03C15

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Full text arXiv 2208.13052: pdf, ps.