2019/10/11 by Schmerl, James H.
#03H15 #FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.1910.05284
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 N5. This theorem implies that every countable nonstandard \mathcal M has an elementary cofinal extension such that Lt(\mathcal N / \mathcal M) ≅ \mathbf N5. It is proved here that if \mathcal M \prec \mathcal N and Lt(\mathcal N / \mathcal M) ≅ \mathbf N5, 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) ≅ \mathbf N5 and \mathcal N^* is neither an end nor a cofinal extension of \mathcal M^*.