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THE PENTAGON AS A SUBSTRUCTURE LATTICE OF MODELS OF PEANO ARITHMETIC
- Part of
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- Journal:
- The Journal of Symbolic Logic , First View
- Published online by Cambridge University Press:
- 29 January 2024, pp. 1-20
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- Article
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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 
${\mathcal M}^*$.
