Jan Kruschewski

About Me

My name is Jan Kruschewski, and I am a mathematician fascinated by the logical foundations of mathematics, essentially the “rules of the game” that all mathematical reasoning follows. I spend much of my time thinking about set theory, the study of collections of objects, and its interaction with higher analogues of computability theory, a field of mathematics that explores the limits of what can be calculated, no matter how powerful our computers or methods become. My research moves between two extremes: very weak systems that capture the absolute essentials of mathematical reasoning, and large cardinal axioms, which reach far into the infinite and describe vast mathematical landscapes. I am particularly interested in how these different perspectives fit together, and what they can teach us about truth, proof, and the infinite.

I earned my PhD at the University of Münster under the supervision of Farmer Schlutzenberg, focusing on Kripke-Platek set theory and its interactions with inner model theory. My work examines the fine-structural and combinatorial properties of weak set theories under stronger axioms, particularly large cardinals. I am also interested in the connections between set theory and computability theory, especially in the context of definability and admissibility. I am currently affiliated with TU Wien, continuing to investigate how weak foundational frameworks interact with strong structural hypotheses in the broader landscape of mathematical logic.

Publications and Preprints

On a Conjecture Regarding the Mouse Order for Weasels with F. Schlutzenberg The Journal of Symbolic Logic, Vol. 90, Issue 1, 2025, DOI, arxiv

We investigate Steel's conjecture in 'The Core Model Iterability Problem', that if W and R are Ω+1-iterable, 1-small weasels, then W≤*R iff there is a club C ⊂ Ω such that for all α∈C, if α is regular, then the cardinal successor of α in W is less or equal than the cardinal successor of α in R . We will show that the conjecture fails, assuming that there is an iterable premouse which models KP and which has a Σ1-Woodin cardinal. On the other hand, we show that assuming there is no transitive model of KP with a Woodin cardinal the conjecture holds. In the course of this we will also show that if M is an iterable admissible premouse with a largest, regular, uncountable cardinal δ, and ℙ is a forcing poset with the δ-c.c. in M, and g is M-generic, but not necessarily Σ1-generic, M[g] is a model of KP. Moreover, if M is such a mouse and T is maximal normal iteration tree on M such that T is non-dropping on its main branch, then its last model is again an iterable admissible premouse with a largest regular and uncountable cardinal. At last, we answer another open question from 'The Core Model Iterability Problem' regarding the S-hull property.

Analysis of HOD for Admissible Structures with F. Schlutzenberg submitted, arxiv

Let n≥1 and assume that there is a Woodin cardinal. Fix x∈ℝ and let α be the least β such that Lβ[x]⊨Σn-KP+∃κ(‘‘κ is inaccessible and κ+ exists"). We adapt the analysis of HODL[x,G] as a strategy mouse to Lαx[x,G] for a cone of reals x. That is, we identify a mouse n-ad and define a class H⊆Lαx[x,G] as a natural analogue of HODL[x,G]⊆L[x,G], and show that H=M∞[Σ0], where M∞ is an iterate of n-ad and Σ0 a fragment of its iteration strategy.

Derived Model Theorem in Weak Set Theories with F. Schlutzenberg in preparation

Let Σ1-KP be Kripke-Platek set theory where the full foundation scheme is replaced with ∈-Foundation and Th∞ be the theory which consists of Σ1-KP and the statement “Vλ exists, where λ is a limit of Woodin cardinals”. Let ThAD be the theory which consists of Σ1-KP, “V= L(R)”, “P(ω) exists”, the Axiom of Determinacy, and the statement “there is a Σ1-elementary substructure which contains the reals”. We will show in the metatheory ZFC that if Con(Th∞) holds, then Con(ThAD) holds. The proof involves a variant of the derived model theorem. The construction of the derived model proceeds by first identifying its Σ1 theory in real parameters via mouse witnesses and then constructing an appropriate term model from this theory. The appropriate version of the key reflection lemma which is used in the classical proof of the derived model theorem is then shown for this term model.

Talks

5th Münster Conference on Inner Model Theory University of Münster On a Conjecture about the Mouse Order for Weasels June 2022

Advances in Set Theory The Herbrew University of Jerusalem On a Conjecture about the Mouse Order for Weasels July 2022 (Poster Presentation)

Arctic Set Theory Workshop 6 Kilpisjärvi, Finland On a Conjecture Regarding the Mouse Order for Weasels February 2023

2nd Irvine Conference on Inner Model Theory UC Irvine Forcing over Admissible Premice July 2023

Vienna Inner Model Theory Conference ESI Vienna Analysis of HOD for Admissible Structures June 2024

Reverse Mathematics and Higher Computability Theory ESI Vienna Analysis of HOD for Admissible Structures July 2025

Logic Colloquium 2025 TU Vienna A Derived Model Theorem in an Admissible Context July 2025

Kruschew.ski

Jan Kruschewski