# Categoricity-like notions for first-order theories

Mateusz Łełyk^{1}

^{1}* University of Warsaw*

The tutorial is devoted to the exposition of recent results revolving around the following vague question:

Are salient foundational theories such as Peano Arithmetic $\sf PA$ or Zermelo-Frankel set theory $\sf ZF$ distinguishable from other theories by virtue of being “more categorical”?

Obviously any theory that can establish the basic truths about addition and multiplication of natural numbers is far from being categorical, in the traditional sense of this term. However, the recent work of Albert Visser, Ali Enayat (tightness and solidity) and Jouko Väänänen (internal categoricity) delivers a bunch of interesting candidates for categoricity-flavoured criteria which can be meaningfully applied to theories of foundations of mathematics. During the tutorial we shall focus mainly on the notions of tightness and solidity. The first part serves as an introduction additionally including the results by Albert Visser on the solidity of $\sf PA$ and by Ali Enayat on the non-tightness of restricted subsystems of $\sf PA$. The content of the second part will be concentrated around the problem of existence of tight/solid subtheories of $\sf PA$ and report the recent results obtained jointly with Leszek Kołodziejczyk and Piotr Gruza. Finally the last part introduces a different perspective on categoricity-like criteria: we experiment with the view that the basic objects to which our criteria are applied are schemes and not theories. This part is based on the recent joint paper with Ali Enayat “Categoricity-like properties in the first-order realm”.

Here is a tentative plan for the tutorials:

**Part 1:**

Introduction of the notions of biinterpretability and tightness.

Proof of tightness of $\sf PA$.

Proof that for every $n$, the set of consequences of $\sf PA$ of complexity at most $\Sigma_n$ is not tight.

Introduction of the notion of solidity.

Proof that the class of solid theories is closed under biinterpretability.

**Part 2:**

Construction of a solid subtheory of $\sf PA$ which does not interpret $\sf PA$.

Construction of a subtheory of $\sf PA$ which is tight but not solid.

**Part 3:**

Introduction of the notion of $g$-solidity for schemes.

Proof that all sufficiently strong theories can be axiomatized by a scheme which is not $g$-solid.

#### References

[1] Albert Visser, Categories of theories and interpretations

[2] Ali Enayat, Variations on a Visserian theme

[3] Ali Enayat, Mateusz Łełyk, Categoricity-like properties in the first-order realm