Theories \(S\) and \(T\) are bi-interpretable if and only if there exist interpretations \(g:S\rhd T\) and \(h:T\rhd S\) such that \(h\circ g\) and \(g\circ h\) are definably isomorphic to identity interpretations (in \(S\) and \(T\) respectively). In some contexts, bi-interpretable theories can be considered equivalent.

It was shown by Visser that Peano Arithmetic has the property that no two distinct extensions of it (in its language) are bi-interpretable. 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, which he called neatness and solidity.

Enayat proved that not only \(\mathrm{PA}\), but also \(\mathrm{ZF}\), \(\mathrm{Z}_{2}\), and \(\mathrm{KM}\) are solid; and on the other hand, that finitely axiomatisable fragments of them are not even tight. Later work by a number of authors (Enayat, Freire, Hamkins, Williams) showed that many natural proper fragments of these theories are also not tight.

Enayat asked whether there are proper solid subtheories (containing basic axioms that depend on the theory) of the theories listed above. We answer this question in the case of \(\mathrm{PA}\) by proving that for every \(n\) there exists a solid theory strictly between \(\mathrm{I}\Sigma_{n}\) and \(\mathrm{PA}\). Furthermore, we can require that the theory does not interpret \(\mathrm{PA}\). We also prove some theorems concerning separations between the above notions, e.g. we show that for every \(n\) there exists a tight but not neat theory strictly between \(\mathrm{I}\Sigma_{n}\) and \(\mathrm{PA}\).

Joint work with Leszek Kołodziejczyk and Mateusz Łełyk.