Modern Czech Logic
in the Philosophy of Mathematics

About

From roughly 1950 onwards, mathematical logic in Prague has begun to flourish. This project investigates the significance and context of the works of three Czech logicians, Ladislav Svante Rieger, Petr Vopěnka, and Petr Hájek, each of whom played a key part in this process and initiated an ongoing tradition.

Reasoning from first principles became a hallmark of the Prague school of formal logic. Working axiomatically enabled deliberate departures from established practices towards entirely new ways of doing and thinking about mathematics. Key topics of interest in the project span various set-theoretic foundations of mathematics, competing views, and pluralism. The work of the three logicians can be fruitfully investigated against the backdrop of finitism. Vopěnka also claimed an affinity with the phenomenological tradition stemming from Husserl's work. Additionally, the early interest in computing, cybernetics, and artificial intelligence branched out into interdisciplinary pursuits.

These creative endeavours do not stand in isolation, but inherit the legacy of Leibniz, Bolzano, Cantor, Husserl, Hilbert, Skolem, Carnap, and many others. They developed alongside other local enterprises, including the pioneering work of Otakar Zich and the phenomenological school of Jan Patočka. The tradition has also informed subsequent research in arithmetic and nonclassical logic. The project advances this wealth of ideas by making it available in archives of formalized mathematical results.

Timeline

• January 2025. Project started
• August 2025. The New Scientist published a feature on ultrafinitism based on the Ultrafinitism Conference, referring to our work on Vopěnka.
• October 2025. We co-organized Horizons: Conference in Honour of Petr Vopěnka at the AKC in Prague.
• November 2025. Aleksi Honkasalo joined the project.
• December 2025. We retrieved an early unpublished manuscript about the Alternative Set Theory from Vopěnka archive at Department of Philosophy, University of West Bohemia in Pilsen.
• April 2026. Jonathan Osinski joined the project.
• November 2026. Roman Kossak's research visit to ICS CAS.

Team

Principal investigator: Zuzana Haniková, Institute of Computer Science, CAS
Co-investigator: Štěpán Holub, Faculty of Mathematics and Physics, Charles University.
Senior member: Vítězslav Švejdar, Institute of Computer Science, CAS
Postdoctoral member: Aleksi Honkasalo, Institute of Computer Science, CAS
Postdoctoral member: Jonathan Osinski, Institute of Computer Science, CAS

Results

• Vítězslav Švejdar: Petr Vopěnka, some early results in set theory. Accepted to The Logica Yearbook.
• Zuzana Haniková: Vopěnka’s Alternative Set Theory as a Framework for Feasible Numbers. Philosophia Mathematica, October 2026.
• Štěpán Holub and Zuzana Haniková: Axiomatic theory of hereditarily finite sets and its fragments. Archive of Formal Proofs, July 2026.
• Zuzana Haniková and Štěpán Holub: Formalization of Fragments of the Theory of Hereditarily Finite Sets. Proc. Tenth Symposium on Working Formal Methods (FROM 2026), EPTCS 452.
• Jonathan Osinski: Compactness beyond choice and HOD. Submitted. ArXiv version.

Talks

• Zuzana Haniková: Two axiomatic frameworks for feasible numbers. Ultrafinitism conference, April 11-13, 2025, Columbia University, New York. (Invited talk.)
• Vítězslav Švejdar: Petr Vopěnka, some early results in set theory. LOGICA 2025, May 12-16, 2025, Hejnice, Czech Republic.
• Štěpán Holub: Vopěnka’s Attempt to Reconstruct the Natural World. Horizons: Conference in Honour of Petr Vopěnka, October 24-25, 2025, Prague, Czech Republic. (Invited talk.)
• Štěpán Holub: Formalization of Set Theory Fragments in Isabelle/HOL. Czech Gathering of Logicians and Beauty of Logic 2026, February 4-6, 2026, Prague, Czech Republic.
• Zuzana Haniková: Formalizing fragments of ZFfin in Isabelle/HOL. Seminar on Applied Mathematical Logic, March 11, 2026, Prague, Czech Republic.
• Zuzana Haniková: Feasibility and nonclassical mathematics. The Mathematical Turn in Philosophy workshop, May 7-8, 2026, University of Warwick, UK. (Invited talk.)
• Jonathan Osinski: Strong Logics for Generating Large Cardinal Notions. LOGICA 2026, May 18-22, 2026, Hejnice, Czech Republic.
• Aleksi Honkasalo: Modal Alternative Set Theory. Logic Colloquium 2026, June 29-July 3, Swansea, UK.
• Štěpán Holub: Formalization of Fragments of the Theory of Hereditarily Finite Sets in Isabelle/HOL. Working Formal Methods Symposium (FROM) 2026, September 15-17, 2026, Timișoara, Romania.
• Zuzana Haniková: Infinite Sets in Bolzano and Vopěnka. Bolzano's work in mathematics and related topics, September 21-22, 2026, Prague, Czech Republic.
• Aleksi Honkasalo: Is Infinity Inherently Vague, and What Does It Mean If It Is? Bolzano's work in mathematics and related topics, September 21-22, 2026, Prague, Czech Republic.

Contacts

• Zuzana Haniková (zuzana at cs.cas.cz)
• Štěpán Holub (holub at karlin.mff.cuni.cz)

Github repository.

This project is funded by the Czech Science Foundation, as a standard project running 2025 to 2027. The principal beneficiary is Institute of Computer Science of the Czech Academy of Sciences (PI Zuzana Haniková). The co-beneficiary is Faculty of Mathematics and Physics, Charles University (coPI Štěpán Holub).

The ICS part of the project enjoys the milieu of the LogICS group at the ICS CAS.

© Zuzana Haniková.
Website design HTML5 UP (Dimension).
Background image adapted from Neoclassicism Enthusiast, CC BY-SA 4.0, via Wikimedia Commons