March 5
Piotr Gruza,
University of Warsaw
Tightness and solidity in fragments of Peano Arithmetic
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 $\text{PA}$, but also $\text{ZF}$, $\text{Z}_{2}$, and $\text{KM}$ are solid; and on the other hand, that finitely axiomatisable fragments of them are not even tight. Later work by a number of authors showed that many natural proper fragments of these theories are also not tight.
Enayat asked whether there are proper solid subtheories (containing some basic axioms that depend on the theory) of the theories listed above. We answer this question in the case of $\text{PA}$ by proving that for every $n$ there exists a solid theory strictly between $\text{I}\Sigma_{n}$ and $\text{PA}$. Furthermore, we can require that the theory does not interpret $\text{PA}$, and that if any true arithmetic sentence is added to it, the theory still does not prove $\text{PA}$.
Joint work with Leszek Kołodziejczyk and Mateusz Łełyk.