Tightness and solidity in fragments of Peano Arithmetic

TL;DR

This paper investigates the properties of tightness, neatness, and solidity in fragments of Peano Arithmetic, establishing the existence of intermediate theories with specific properties and exploring their interpretability and relationships.

Contribution

It proves the existence of proper solid and tight but not neat theories between fragments of Peano Arithmetic, and examines their interpretability and related properties.

Findings

Existence of solid theories between IΣₙ and PA.

Existence of tight but not neat theories between IΣₙ and PA.

Solid subtheories of PA can be non-interpreting PA.

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 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 $\mathrm{PA}$, but also $\mathrm{ZF}$ and $\mathrm{Z}_2$ 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 $\mathrm{PA}$ by proving that for every $n$, there exist both a solid theory and a tight but not neat theory strictly between $\mathrm{I}\Sigma_{n}$ and $\mathrm{PA}$. Moreover, the solid subtheories of $\mathrm{PA}$ can be required to be unable to interpret $\mathrm{PA}$. We also obtain some other separations between properties related to tightness, for example by giving an example of a sequential theory that is neat but not semantically tight in the sense of Freire and Hamkins.