This project studies the structural complexity of foundational theories and investigates when a theory has an intended model. We use advances in Scott analysis and new techniques to understand how intendedness arises, and how model‑theoretic complexity can be detected by infinitary logic.

Research questions

  • Do foundational theories have a structurally simplest model, and is it the intended one?
  • Can intendedness be characterized by other properties of the theory?
  • How do Scott functions reflect model‑theoretic and descriptive‑set‑theoretic complexity?

Approach

We combine tools from Scott analysis with methods from the foundations of mathematics. The project is interdisciplinary and international, with regular collaboration between the teams in Vienna and Warsaw.

Funding

This project is funded by the Austrian Science Fund (FWF) and by the Polish Science Foundation (NCN). See the FWF project page

Participants

  • Dino Rossegger (Technische Universität Wien)
  • Mateusz Łełyk (University of Warsaw)

Publications

  1. Dichotomy results for classes of countable graphsDOIarXiv
    Vittorio Cipriani, Ekaterina Fokina, Matthew Harrison-Trainor, Liling Ko, and Dino Rossegger
    submitted for publication (2025)
  2. Uniformity in learning structuresDOIarXiv
    Vittorio Cipriani and Dino Rossegger
    submitted for publication (2025)
  3. Classifying the complexity of models of arithmeticDOIarXiv
    David Gonzalez, Mateusz Łełyk, Dino Rossegger, and Patryk Szlufik
    submitted for publication (2025)