The Pentagon as a Substructure Lattice of Models of Peano Arithmetic

TL;DR

This paper explores the structure of models of Peano Arithmetic, showing how the pentagon lattice can be realized as a substructure lattice in certain extensions, and clarifies conditions under which these extensions are end or cofinal.

Contribution

It proves that if the lattice of an extension is a pentagon, then the extension is either an end or cofinal extension, but also provides counterexamples in nonstandard models.

Findings

Pentagon lattice can be realized as a substructure lattice of models of PA.

Extensions with pentagon lattice are either end or cofinal extensions.

Counterexamples exist where the lattice is pentagon but the extension is neither end nor cofinal.

Abstract

Wilke proved in 1977 that every countable model ${\mathcal M}$ of Peano Arithmetic has an elementary end extension ${\mathcal N}$ such that the interstructure lattice Lt(${\mathcal N} / {\mathcal M}$) is the pentagon lattice ${\mathbf N}_5$. This theorem implies that every countable nonstandard $\mathcal M$ has an elementary cofinal extension such that Lt(${\mathcal N} / {\mathcal M}) \cong {\mathbf N}_5$. It is proved here that if ${\mathcal M} \prec {\mathcal N}$ and Lt(${\mathcal N} / {\mathcal M}) \cong {\mathbf N}_5$, then ${\mathcal N}$ is either an end or a cofinal extension of ${\mathcal M}$. In contrast, there are ${\mathcal M}^* \prec {\mathcal N}^* \models {\mathsf PA}^*$ such that Lt(${\mathcal N} / {\mathcal M}) \cong {\mathbf N}_5$ and ${\mathcal N}^*$ is neither an end nor a cofinal extension of ${\mathcal M}^*$.