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10Structural Properties of the Stable CoreJournal of Symbolic Logic 88 (3): 889-918. 2023.The stable core, an inner model of the form $\langle L[S],\in, S\rangle $ for a simply definable predicate S, was introduced by the first author in [8], where he showed that V is a class forcing extension of its stable core. We study the structural properties of the stable core and its interactions with large cardinals. We show that the $\operatorname {GCH} $ can fail at all regular cardinals in the stable core, that the stable core can have a discrete proper class of measurable cardinals, but t…Read more
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15An undecidable extension of Morley's theorem on the number of countable modelsAnnals of Pure and Applied Logic 174 (9): 103317. 2023.
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37Mice with finitely many Woodin cardinals from optimal determinacy hypothesesJournal of Mathematical Logic 20 (Supp01): 1950013. 2020.We prove the following result which is due to the third author. Let [Formula: see text]. If [Formula: see text] determinacy and [Formula: see text] determinacy both hold true and there is no [Formula: see text]-definable [Formula: see text]-sequence of pairwise distinct reals, then [Formula: see text] exists and is [Formula: see text]-iterable. The proof yields that [Formula: see text] determinacy implies that [Formula: see text] exists and is [Formula: see text]-iterable for all reals [Formula:…Read more
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9Constructing wadge classesBulletin of Symbolic Logic 28 (2): 207-257. 2022.We show that, assuming the Axiom of Determinacy, every non-selfdual Wadge class can be constructed by starting with those of level $\omega _1$ and iteratively applying the operations of expansion and separated differences. The proof is essentially due to Louveau, and it yields at the same time a new proof of a theorem of Van Wesep. The exposition is self-contained, except for facts from classical descriptive set theory.
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9In inner models with Woodin cardinalsJournal of Symbolic Logic 86 (3): 871-896. 2021.We analyze the hereditarily ordinal definable sets $\operatorname {HOD} $ in $M_n[g]$ for a Turing cone of reals x, where $M_n$ is the canonical inner model with n Woodin cardinals build over x and g is generic over $M_n$ for the Lévy collapse up to its bottom inaccessible cardinal. We prove that assuming $\boldsymbol \Pi ^1_{n+2}$ -determinacy, for a Turing cone of reals x, $\operatorname {HOD} ^{M_n[g]} = M_n,$ where $\mathcal {M}_{\infty }$ is a direct limit of iterates of $M_{n+1}$, $\delta …Read more
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15Closure properties of measurable ultrapowersJournal of Symbolic Logic 86 (2): 762-784. 2021.We study closure properties of measurable ultrapowers with respect to Hamkin's notion of freshness and show that the extent of these properties highly depends on the combinatorial properties of the underlying model of set theory. In one direction, a result of Sakai shows that, by collapsing a strongly compact cardinal to become the double successor of a measurable cardinal, it is possible to obtain a model of set theory in which such ultrapowers possess the strongest possible closure properties.…Read more
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15Infinite decreasing chains in the Mitchell orderArchive for Mathematical Logic 60 (6): 771-781. 2021.It is known that the behavior of the Mitchell order substantially changes at the level of rank-to-rank extenders, as it ceases to be well-founded. While the possible partial order structure of the Mitchell order below rank-to-rank extenders is considered to be well understood, little is known about the structure in the ill-founded case. The purpose of the paper is to make a first step in understanding this case, by studying the extent to which the Mitchell order can be ill-founded. Our main resu…Read more
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25Long games and σ-projective setsAnnals of Pure and Applied Logic 172 (4): 102939. 2021.We prove a number of results on the determinacy of σ-projective sets of reals, i.e., those belonging to the smallest pointclass containing the open sets and closed under complements, countable unions, and projections. We first prove the equivalence between σ-projective determinacy and the determinacy of certain classes of games of variable length <ω^2 (Theorem 2.4). We then give an elementary proof of the determinacy of σ-projective sets from optimal large-cardinal hypotheses (Theorem 4.4). Fina…Read more
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34Projective Games on the RealsNotre Dame Journal of Formal Logic 61 (4): 573-589. 2020.Let Mn♯ denote the minimal active iterable extender model which has n Woodin cardinals and contains all reals, if it exists, in which case we denote by Mn the class-sized model obtained by iterating the topmost measure of Mn class-many times. We characterize the sets of reals which are Σ1-definable from R over Mn, under the assumption that projective games on reals are determined:1. for even n, Σ1Mn=⅁RΠn+11;2. for odd n, Σ1Mn=⅁RΣn+11.This generalizes a theorem of Martin and Steel for L, that is,…Read more
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25J. Steel, PFA implies_ AD _L_(ℝ). _ _The Journal of Symbolic Logic_ _, vol. 70 (2005), no. 4, pp. 1255–1296. - G. Sargsyan, _Nontame mouse from the failure of square at a singular strong limit cardinal_. _ _Journal of Mathematical Logic_ _, vol. 14 (2014), 1450003 (47 pages). - G. Sargsyan, _Covering with universally Baire operators_. _ _Advances in Mathematics_ _, vol. 268 (2015), pp. 603–665. - N. Trang, _PFA and guessing models_. _ _Israel Journal of Mathematics_ , vol. 215 (2016), pp. 607–667 (review)Bulletin of Symbolic Logic 26 (1): 89-92. 2020.
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40Mice with finitely many Woodin cardinals from optimal determinacy hypothesesJournal of Mathematical Logic 20 (Supp01): 1950013. 2020.We prove the following result which is due to the third author. Let [Formula: see text]. If [Formula: see text] determinacy and [Formula: see text] determinacy both hold true and there is no [Formula: see text]-definable [Formula: see text]-sequence of pairwise distinct reals, then [Formula: see text] exists and is [Formula: see text]-iterable. The proof yields that [Formula: see text] determinacy implies that [Formula: see text] exists and is [Formula: see text]-iterable for all reals [Formula:…Read more
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18Every zero-dimensional homogeneous space is strongly homogeneous under determinacyJournal of Mathematical Logic 20 (3): 2050015. 2020.All spaces are assumed to be separable and metrizable. We show that, assuming the Axiom of Determinacy, every zero-dimensional homogeneous space is strongly homogeneous (i.e. all its non-empty clopen subspaces are homeomorphic), with the trivial exception of locally compact spaces. In fact, we obtain a more general result on the uniqueness of zero-dimensional homogeneous spaces which generate a given Wadge class. This extends work of van Engelen (who obtained the corresponding results for Borel …Read more
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46The consistency strength of long projective determinacyJournal of Symbolic Logic 85 (1): 338-366. 2019.We determine the consistency strength of determinacy for projective games of length ω^2. Our main theorem is that $\Pi _{n + 1}^1$-determinacy for games of length ω^2 implies the existence of a model of set theory with ω + n Woodin cardinals. In a first step, we show that this hypothesis implies that there is a countable set of reals A such that M_n(A), the canonical inner model for n Woodin cardinals constructed over A, satisfies $A = R$ and the Axiom of Determinacy. Then we argue how to obtain…Read more
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9The axiom of determinacy implies dependent choice in miceMathematical Logic Quarterly 65 (3): 370-375. 2019.We show that the Axiom of Dependent Choice,, holds in countably iterable, passive premice constructed over their reals which satisfy the Axiom of Determinacy,, in a background universe. This generalizes an argument of Kechris for using Steel's analysis of scales in mice. In particular, we show that for any and any countable set of reals A so that and, we have that.
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Ludwig Maximilians Universität, MünchenFaculty of Philosophy, Philosophy of Science and Study of ReligionTeaching Assistant (Part-time)
Hochschule Für Philosophie In München
Alumnus, 2016
München, Bavaria, Germany