Tightness and solidity in fragments of Peano Arithmetic

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.

Original Section 4 from v1 split into Section 4, containing the basic form of the main result, and Section 5, containing refinements. Examples of proper solid subtheories of ZF included (Proposition 7.1). Minor textual and bibliographic updates throughout