五边形作为皮亚诺算术模型的子结构网格

The Journal of Symbolic Logic Pub Date : 2024-01-29 DOI:10.1017/jsl.2024.6

JAMES H. SCHMERL

引用次数: 0

摘要

威尔基（Wilkie）在1977年证明了每一个皮亚诺算术的可数模型${/mathcal M}$都有一个基本末端扩展${/mathcal N}$，使得结构间网格$operatorname {\mathrm {Lt}}({\mathcal N} / {\mathcal M})$ 是五边形网格${/mathbf N}_5$。这个定理意味着，每一个可数非标准 ${\mathcal M}$ 都有一个基本同尾扩展 ${\mathcal N}$ ，使得 $\operatorname {mathrm {Lt}}({\mathcal N} / {\mathcal M}) \cong {\mathbf N}_5$.这里证明，只要 ${\mathcal M}\就可以证明\且 $\operatorname {\mathrm {Lt}}({\mathcal N} / {\mathcal M}) \cong {\mathbf N}_5$ 时，那么 ${mathcal N}$ 一定是 ${mathcal M}$ 的末尾或共末尾扩展。相反地有 ${\mathcal M}^* \prec {\mathcal N}^* \models \mathsf {PA}^*$ 使得 $\operatorname {mathrm {Lt}}({\mathcal N}^* / {\mathcal M}^*) \cong {\mathbf N}_5$ 并且 ${\mathcal N}^*$ 既不是末端也不是 ${\mathcal M}^*$ 的同末端扩展。

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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 ${\mathcal M}^*$.

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The Journal of Symbolic Logic

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