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.
Categories: math.LO