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arXiv · 2208.01697

The structural complexity of models of arithmetic

Antonio Montalbán·Dino Rossegger
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Abstract

We calculate the possible Scott ranks of countable models of Peano arithmetic. We show that no non-standard model can have Scott rank less than $\omega$ and that non-standard models of true arithmetic must have Scott rank greater than $\omega$. Other than that there are no restrictions. By giving a reduction via $\Delta^{\mathrm{in}}_{1}$ bi-interpretability from the class of linear orderings to the canonical structural $\omega$-jump of models of an arbitrary completion $T$ of $\mathrm{PA}$ we show that every countable ordinal $\alpha>\omega$ is realized as the Scott rank of a model of $T$.

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Antonio Montalbán, Dino Rossegger. 2022-08-02. The structural complexity of models of arithmetic. https://arxiv.org/abs/2208.01697

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