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THE LATTICE PROBLEM FOR MODELS OF $\mathsf {PA}$

Part of: Nonstandard models Model theory

Published online by Cambridge University Press:  13 January 2025

ATHAR ABDUL-QUADER Open the ORCID record for ATHAR ABDUL-QUADER [Opens in a new window]  and
ROMAN KOSSAK Open the ORCID record for ROMAN KOSSAK [Opens in a new window]
ATHAR ABDUL-QUADER*
Affiliation:
SCHOOL OF NATURAL AND SOCIAL SCIENCES PURCHASE COLLEGE, SUNY PURCHASE, NY 10577, USA
ROMAN KOSSAK
Affiliation:
DEPARTMENT OF MATHEMATICS, THE GRADUATE CENTER CITY UNIVERSITY OF NEW YORK NEW YORK, NY 10016, USA E-mail: rkossak@gc.cuny.edu
*
E-mail: athar.abdulquader@purchase.edu
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Abstract

The lattice problem for models of Peano Arithmetic ($\mathsf {PA}$) is to determine which lattices can be represented as lattices of elementary submodels of a model of $\mathsf {PA}$, or, in greater generality, for a given model $\mathcal {M}$, which lattices can be represented as interstructure lattices of elementary submodels $\mathcal {K}$ of an elementary extension $\mathcal {N}$ such that $\mathcal {M}\preccurlyeq \mathcal {K}\preccurlyeq \mathcal {N}$. The problem has been studied for the last 60 years and the results and their proofs show an interesting interplay between the model theory of PA, Ramsey style combinatorics, lattice representation theory, and elementary number theory. We present a survey of the most important results together with a detailed analysis of some special cases to explain and motivate a technique developed by James Schmerl for constructing elementary extensions with prescribed interstructure lattices. The last section is devoted to a discussion of lesser-known results about lattices of elementary submodels of countable recursively saturated models of PA.

Keywords

models of arithmeticPeano Arithmeticlatticeslattices of elementary substructures

MSC classification

Primary: 03C62: Models of arithmetic and set theory 03H15: Nonstandard models of arithmetic

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Type
Article
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Bulletin of Symbolic Logic , First View , pp. 1 - 28
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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