THE PENTAGON AS A SUBSTRUCTURE LATTICE OF MODELS OF PEANO ARITHMETIC

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