My research interests are mainly in determinacy and inner model theory, but I am also interested in wider areas of set theory that are potentially related to determinacy.
My research interests are mainly in determinacy and inner model theory, but I am also interested in wider areas of set theory that are potentially related to determinacy.
Determinacy of long games just beyond fixed countable length (with S. Müller).
Equivalence between mice and determinacy of games of fixed countable length [tentative title] (with J. P. Aguilera).
Separating Maximality Principles (with A. Lietz). PDF
Generic absoluteness revisited (with S. Fuchino and F. Parente), submitted. arXiv.
The present paper is concerned with the relation between recurrence axioms and Laver-generic large cardinal axioms in light of principles of generic absoluteness and the Ground Axiom. Viale proved that Martin's Maximum++ together with the assumption that there are class many Woodin cardinals implies H(ℵ2)^V ≺_{Σ2} H(ℵ2)^{V[G]} for a generic G on any stationary preserving P which also preserves Bounded Martin's Maximum. We show that a similar but more general conclusion follows from each of (P,H(κ))_{Σ2}-RcA+ and the existence of the tightly P-Laver-generically huge cardinal. While the assumptions of Viale's Theorem are compatible with the Ground Axiom, we show that the assumptions of our theorems imply the negation of the Ground Axiom. This fact is used to show that fragments of Recurrence Axiom (P,H(κ))_Γ-RcA+ can be different from the corresponding fragments of Maximality Principle MP(P,H(κ))_Γ for Γ=Π2, Σ2.
Determinacy in the Chang model (with G. Sargsyan), accepted to Proceedings of the AMS. arXiv.
Assuming the existence of a certain hod pair with a Woodin cardinal that is a limit of Woodin cardinals, we show that the Chang model satisfies AD^+ in any set generic extensions.
Chang models over derived models with supercompact measures (with S. Müller and G. Sargsyan), Journal of Mathematical Logic, Online ready: 2550007. DOI, arXiv.
Based on earlier work of the third author, we construct a Chang-type model with supercompact measures extending a derived model of a given hod mouse with a regular cardinal δ that is both a limit of Woodin cardinals and a limit of <δ-strong cardinals. The existence of such a hod mouse is consistent relative to a Woodin cardinal that is a limit of Woodin cardinals. We argue that our Chang-type model satisfies AD_R + Θ is regular + ω1 is <δ∞-supercompact for some regular cardinal δ∞>Θ. This complements Woodin's generalized Chang model, which satisfies AD_R + ω1 is supercompact, assuming a proper class of Woodin cardinals that are limits of Woodin cardinals.
On ω-strongly measurable cardinals in Pmax extensions (with N. Aksornthong, J. Holland, and G. Sargsyan), Journal of Mathematical Logic 25, issue 2 (2025): 2450018. DOI, arXiv.
We show that in the Pmax extension of a certain Chang-type model of determinacy, if κ∈{ω1,ω2,ω3}, then the restriction of the club filter on κ∩Cof(ω) to HOD is an ultrafilter in HOD. This answers Question 4.11 of the paper "On ω-Strongly Measurable Cardinals" raised by Ben-Neria and Hayut.
On the derived models in self-iterable universes (with G. Sargsyan), Proceedings of the AMS150, no. 3 (2022): 1321-1329. DOI, arXiv.
We show that if the universe is self-iterable and κ is an inaccessible limit of Woodin cardinal then "AD_R + Θ is regular" holds in the derived model at κ. The proof is fine-structure free, and only assumes basic knowledge of iteration trees and iteration strategies. Our proof can be viewed as the fine-structure free version of the well-known fact that "AD_R + Θ is regular" is true in the derived models of hod mice that have inaccessible limit of Woodin cardinals. However, the proof uses a different set of ideas and is more general.
Compactness of ω1 and strong axioms of determinacy, The Bulletin of Symbolic Logic 30, issue 2 (2024): 279-282. DOI.
In this review, we summarize the recent progress in the study of super/strong compactness of ω1 and strong forms of determinacy.
Correction: The line (4-1) should be "ZFC + There is a Woodin limit of Woodin cardinals."
Additional note: Recently, Blue, Sargsyan and Larson announced in their preprint on Nairian models that the consistency strength of "AD_R + Θ is regular + ω1 is supercompact" is strictly weaker than the existence of a Woodin limit of Woodin cardinals, which refutes the conjecture of Ikegami and Trang mentioned in this review.
Personal notes on Mitchell–Steel mice. (available by request)
Introduction to iteration trees. Handwritten note of my talks at Inner Model Theory Seminar at TU Wien. PDF
This note was written for a reading seminar of Steel's lecture note, "Introduction to Iterated Ultrapowers." Starting from the definition of iteration trees, I give a proof sketch of the branch existence theorem for iteration trees of length omega.
Handwritten notes of the lectures of Pmax Tutorial by Paul Larson at TU Wien. Week 1: PDF Week 2: PDF
Week 1: Basic Analysis of the Pmax forcing. For example, it was shown that in any Pmax extension of an AD^+ model, the nonstationary ideal on omega_1 is saturated, delta^1_2 is omega_2, and the continuum is omega_2.
Week 2: Advanced results of the Pmax forcing. The following topics are included: Forcing MM^++(c) over models of "AD_R + Theta is regular." Overview of Larson and Sargsyan's result on failures of square principles in Pmax extensions. Some comments on Woodin's result on the equivalence of (*)^+ and (*)^++.
Chang-type models of determinacy. PDF
These slides were used at Logic Colloquium 2023 on June 9, 2023. The talk is based on several joint works with N. Aksornthong, J. Holland, S. Müller and G. Sargsyan.
These are my slides used at Kobe Set Theory Seminar on May 15 & 22, 2023. This is joint work with S. Müller and G. Sargsyan. (May 31, 2023: the proof given in Part II contains an error. We correct the error in the preprint.)
Determinacy in the Chang model from a hod pair. PDF
This is a revised version of my slides used at Core Model Seminar on January 31, 2023. This is joint work with G. Sargsyan.
Chang models over derived models. PDF
This is a revised version of my slides used at Core Model Seminar on January 24, 2023. In the slides, I give a proof outline of the main theorem in Sargsyan's paper, "Covering with Chang models over derived models."