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Abstract:It was shown by Visser that Peano Arithmetic has the property that any two bi-interpretable extensions of it (in the same language) are equivalent. Enayat proposed to refer to this property of a theory as \emph{tightness} and to carry out a more systematic study of tightness and its stronger variants that he called neatness and solidity. Enayat proved that not only $\mathsf{PA}$, but also $\mathsf{ZF}$ and $\mathsf{Z}_2$ are solid. On the other hand, it was shown in later work by a number of authors that many natural proper fragments of those theories are not even tight. Enayat asked whether there is a proper solid subtheory of the theories listed above. We answer that question in the case of $\mathsf{PA}$ by proving that for every $n$, there exist both a solid theory and a tight but not neat theory strictly between $\mathsf{I}\Sigma_n$ and $\mathsf{PA}$. Moreover, the solid subtheories of $\mathsf{PA}$ can be required to be unable to interpret $\mathsf{PA}$. We also provide simple examples of proper solid subtheories of $\mathsf{ZF}$ and $\mathsf{Z}_2$, as well as further separations between properties related to tightness, including an example of a sequential theory that is neat but not semantically tight in the sense of Freire and Hamkins.

Submission history

From: Mateusz Łełyk [view email]
[v1] Tue, 9 Dec 2025 21:01:56 UTC (64 KB)
[v2] Wed, 24 Jun 2026 20:19:52 UTC (59 KB)