Albert Visser
University of Utrecht
Introduction to Interpretations
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Lecture 1: Basics
Lecture 2: Categories and Sameness
Lecture 3: The Interpretation Existence Lemma
Lecture 4: The Smullyan Bootstrap
There could be many introductions to interpretations, so the tutorial will, in some measure, reflect my personal interests.
We discuss two connected questions. When are two interpretations the same? When are two theories the same? We illustrate how different notions of sameness of interpretation lead to different notions of sameness of theory.
A central result of logic is the Model Existence Lemma. We show how this result has a theory-internal shadow: the Interpretation Existence Lemma. Roughly, if theory T proves the consistency of theory U, then T interprets U. We consider some applications of this fundamental result.
We enrich theories for various applications like the incompleteness theorems. A bootstrap for enrichment often takes the form of a series of interpretations. We illustrate this methodology by zooming in on a specific bootstrap that is aimed at adding sequences to a theory. The first step of our bootstrap is the Smullyan interpretation of binary strings. The second step builds sequences from strings.

Mateusz Łełyk
University of Warsaw
Categoricity first(-orderized)
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Very informally, a way of "first-orderizing" a concept consists in requiring all the sets occuring in its definition to be definable. Since a definable model is essentially an interpretation, by first-orderizing categoricity one arrives at the following definition: theory T is interpretation categorical iff every self-interpretation of T is T-provably isomorphic to the identity interpretation. This definition was originally proposed by Mirko Engler and Benjamin Zayton (they call it rigidity). In this series of lectures, starting from interpretation categoricity, in each lecture we discuss a different categoricity-like notion and explain the state of art w.r.t. which foundational theories (or schemes) satisfy it.
1. Interpretation categoricity. We plan to prove the interpretation categoricity of the true arithmetic and generalize this result. We observe that sound, r.e. sequential theories cannot be interpretation categorical.
2. Retraction categoricity. We weaken interpretation categoricity by requiring only that any self-retraction (self-interpretation that admit left inverses) of T is trivial. By adapting Visser's result we show that PA is retraction categorical. We give an unpublished proof due to Leszek Kołodziejczyk that each I\Sigma_n admits a non-trivial self-retraction.
3. Solidity. Strengthening retraction categoricity leads to solidity. We give a proof of solidity of ZF (due to Ali Enayat) and at least sketch the construction of a proper solid subtheory of PA (due to Piotr Gruza, Leszek Kołodziejczyk and myself, "Tightness and solidity in fragments of Peano Arithmetic".
4. Explaining patterns: a scheme perspective. We explain a fruitful perspective on categoricity, which is obtained by shifting attention from theories to schemes. This is based on our recent draft "Definiteness properties of first-order schemes", joint with Piotr Gruza).

Isabel Oitavem
Universidade Nova de Lisboa
Recursion and Complexity
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A central goal in computer science is to understand why some problems are easy to solve while others seem hopelessly difficult — the famous P versus NP problem is the best-known example. Early approaches measured difficulty by the time or memory a computer needs, which was perceived as missing the mathematics of the problems themselves. A major shift occurred in 1974 when Ron Fagin showed that NP corresponds exactly to problems expressible in a particular logical language. This revealed a powerful idea: the complexity of a problem is connected to the expressive power needed to describe it, not just the computational resources needed to solve it. Since then, many machine-independent characterisations of complexity classes have been developed.
In this course, we explore recursion-theoretic approaches to the study of complexity classes. This perspective reveals deep connections between complexity theory and mathematical tools such as mathematical induction, proof theory, and type systems. We will examine fundamental complexity classes, including P and NP, alongside others. The course is intended to be self-contained, providing the necessary background and developing the relevant concepts.

(Tentative) Schedule for Both Days (8-9 October 2026)

• 10:15–11:00 – Albert Visser
• 11:00–11:30 – Coffee Break
• 11:30–12:15 – Albert Visser
• 12:15–13:30 – Lunch
• 13:30–14:15 – Mateusz Łełyk
• 14:15–14:25 – Short Break
• 14:25–15:10 – Mateusz Łełyk
• 15:10–15:40 – Coffee Break
• 15:40–16:25 – Isabel Oitavem
• 16:25–16:30 – Short Break
• 16:30–17:15 – Isabel Oitavem

The workshop is free of charge and open to all interested participants. No registration is required to attend the workshop.

This workshop is preceded by the workshop Metamathematical Miscellanea which will take place in Lugano the 5th and 6th October. Anyone is welcome to attend this workshop as well. For more information, please consult this website.