arXiv subjects

Grigor Sargsyan

Publications and source records attributed to Grigor Sargsyan.

At least 19 recordsLinked to original sources

arXiv2026

Future directions in nuclear $β$ decay at FRIB and beyond

Motivated by the opportunities presented for studies relevant to nuclear structure, astrophysics, and fundamental symmetries with nuclear $β$ decay, the Facility for Rare Isotope Beams (FRIB) Theory Alliance topical program ``Future Directions in Nuclear $β$ Decays at FRIB'' was held in September of 2025. This white paper summarizes the main points of discussion over the two-week program, and it aims to provide a snapshot of the current status of the field while also highlighting important questions and opportunities for future work. We provide an overview of the experimental tools and techniques that enable modern $β$ decay studies, discuss the current state of nuclear many-body approaches used to study $β$ decays, and highlight the important science questions that can be addressed by weak decays.

nucl-th↗

arXiv2026

Ultrapowers of determinacy models as iteration trees on HOD

In the 1990s, Steel and Woodin showed that under large cardinal hypotheses, the HOD of $L(\mathbb R)$ admits a fine-structural analysis. Although this theorem sheds light on various problems in descriptive set theory, the fine-structural representations of many fundamental objects of determinacy theory are still unknown. For example, Woodin asked whether the ultrapower of HOD by the closed unbounded filter on $ω_1$ is given by an iteration tree on HOD according to its fine-structural extender sequence and canonical iteration strategy. In this paper, we give a positive answer to Woodin's question, not only for the closed unbounded filter but for any ultrafilter on an ordinal. The key tool that enables the solution of Woodin's problem is a recent advance in inner model theory: the Steel--Schlutzenberg theory of normalizing iteration trees, which allows us to represent HOD and its ultrapowers as normal iterates of a single countable mouse. Despite our results, the precise structure of the iteration trees that lead from HOD into its ultrapowers remains a mystery.

math.LO↗

arXiv2026

Unreachability of Inductive-Like Pointclasses in $L(\mathbb{R})$

Hjorth proved from $ZF + AD + DC$ that there is no sequence of distinct $Σ^1_2$ sets of length $δ^1_2$. Sargsyan extended Hjorth's technique to show there is no sequence of distinct $Σ^1_{2n}$ sets of length $δ^1_{2n}$. Sargsyan conjectured an analogous property is true for any regular Suslin pointclass in $L(R)$ -- i.e. if $κ$ is a regular Suslin cardinal in $L(R)$, then there is no sequence of distinct $κ$-Suslin sets of length $κ^+$ in $L(R)$. We prove this in the case that the pointclass $S(κ)$ is inductive-like.

math.LO↗

arXiv2026

The failure of square at all uncountable cardinals is weaker than a Woodin limit of Woodin cardinals

We force the Axiom of Choice over the least initial segment of a Nairian model satisfying ZF. In the forcing extension, square_kappa fails at all uncountable cardinals kappa, and every regular cardinal is omega-strongly measurable in HOD, as witnessed by the omega-club filter. Thus the failure of square everywhere is within the current reach of inner model theory, and the HOD Hypothesis is not provable in ZFC.

math.LO↗

arXiv2025

Partial Tower Sealing

The main result of this paper shows that a weak form of Tower Sealing holds in a generic extension of hod mice with a strong cardinal and a proper class of Woodin cardinals. We show Tower Sealing fails in such extensions in general. We show that this weak form of Tower Sealing (called Partial Tower Sealing) implies Sealing and that its consistency strength is below that of ZFC + there is a Woodin limit of Woodin cardinals.

math.LO↗

arXiv2025

The exact strength of generic absoluteness for the universally Baire sets

A set of reals is \textit{universally Baire} if all of its continuous preimages in topological spaces have the Baire property. $\sf{Sealing}$ is a type of generic absoluteness condition introduced by Woodin that asserts in strong terms that the theory of the universally Baire sets cannot be changed by forcing. The $\sf{Largest\ Suslin\ Axiom}$ ($\sf{LSA}$) is a determinacy axiom isolated by Woodin. It asserts that the largest Suslin cardinal is inaccessible for ordinal definable bijections. Let $\sf{LSA-over-uB}$ be the statement that in all (set) generic extensions there is a model of $\sf{LSA}$ whose Suslin, co-Suslin sets are the universally Baire sets. We show that over some mild large cardinal theory, $\sf{Sealing}$ is equiconsistent with $\sf{LSA-over-uB}$. In fact, we isolate an exact large cardinal theory that is equiconsistent with both (see \rdef{dfn:hod_pm}). As a consequence, we obtain that $\sf{Sealing}$ is weaker than the theory $``\sf{ZFC} + $there is a Woodin cardinal which is a limit of Woodin cardinals". A variation of $\sf{Sealing}$, called $\sf{Tower \ Sealing}$, is also shown to be equiconsistent with $\sf{Sealing}$ over the same large cardinal theory. The result is proven via Woodin's $\sf{Core\ Model\ Induction}$ technique, and is essentially the ultimate equiconsistency that can be proven via the current interpretation of $\sf{CMI}$ as explained in the paper.

math.LO↗

arXiv2025

On $ω$-strongly measurable cardinals in $\mathbb{P}_{\max}$ extensions

We show that in the $\mathbb{P}_{\max}$ extension of a certain Chang-type model of determinacy, if $κ\in\{ω_1, ω_2, ω_3\}$, then the restriction of the club filter on $κ\cap\mathrm{Cof}(ω)$ to HOD is an ultrafilter in HOD. This answers Question 4.11 of [BNH23] raised by Ben-Neria and Hayut.

math.LO↗

arXiv2025

Varsovian models II

Assume the existence of sufficent large cardinals. Let $M_{\mathrm{sw}n}$ be the minimal iterable proper class $L[E]$ model satisfying "there are $δ_0<κ_0<\ldots<δ_{n-1}<κ_{n-1}$ such that the $δ_i$ are Woodin cardinals and the $κ_i$ are strong cardinals". Let $M=M_{\mathrm{sw}2}$. We identify an inner model $\mathscr{V}_2^M$ of $M$, which is a proper class model satisfying "there are 2 Woodin cardinals", and is iterable both in $V$ and in $M$, and closed under its own iteration strategy. The construction also yields significant information about the extent to which $M$ knows its own iteration strategy. We characterize the universe of $\mathscr{V}_2^M$ as the mantle and the least ground of $M$, and as $\mathrm{HOD}^{M[G]}$ for $G\subseteq\mathrm{Coll}(ω,λ)$ being $M$-generic with $λ$ sufficiently large. These results correspond to facts already known for $M_{\mathrm{sw}1}$, and the proofs are an elaboration of those, but there are substantial new issues and new methods used to handle them.

math.LO↗

arXiv2025

Towards a generic absoluteness theorem for Chang models

Let $Γ^\infty$ be the set of all universally Baire sets of reals. Inspired by recent work of the second author and Nam Trang, we introduce a new technique for establishing generic absoluteness results for models containing $Γ^\infty$. Our main technical tool is an iteration that realizes $Γ^\infty$ as the sets of reals in a derived model of some iterate of $V$. We show, from a supercompact cardinal $κ$ and a proper class of Woodin cardinals, that whenever $g \subseteq Col(ω, 2^{2^κ})$ is $V$-generic and $h$ is $V[g]$-generic for some poset $\mathbb{P}\in V[g]$, there is an elementary embedding $j: V\rightarrow M$ such that $j(κ)=ω_1^{V[g*h]}$ and $L(Γ^\infty, \mathbb{R})$ as computed in $V[g*h]$ is a derived model of $M$ at $j(κ)$. As a corollary we obtain that $\mathsf{Sealing}$ holds in $V[g]$, which was previously demonstrated by Woodin using the stationary tower forcing. Also, using a theorem of Woodin, we conclude that the derived model of $V$ at $κ$ satisfies $\mathsf{AD}_{\mathbb{R}}+``Θ$ is a regular cardinal". Inspired by core model induction, we introduce the definable powerset $\mathcal{A}^\infty$ of $Γ^\infty$ and use our derived model representation mentioned above to show that the theory of $L(\mathcal{A}^\infty)$ cannot be changed by forcing. Working in a different direction, we also show that the theory of $L(Γ^\infty, \mathbb{R})[\mathcal{C}]$, where $\mathcal{C}$ is the club filter on $\wp_{ω_1}(Γ^\infty)$, cannot be changed by forcing. Proving the two aforementioned results is the first step towards showing that the theory of $L(Ord^ω, Γ^\infty, \mathbb{R})([μ_α: α\in Ord])$, where $μ_α$ is the club filter on $\wp_{ω_1}(α)$, cannot be changed by forcing.

math.LO↗

arXiv2025

Hjorth's reflection argument

Hjorth, assuming ${\sf{AD+ZF+DC}}$, showed that there is no sequence of length $ω_2$ consisting of distinct $Σ^1_2$-sets. We show that the same theory implies that for $n\geq 0$, there is no sequence of length $δ^1_{2n+2}$ consisting of distinct $Σ^1_{2n+2}$ sets. The theorem settles Question 30.21 of Kanamori, which was also conjectured by Kechris.

math.LO↗

arXiv2025

Generic Generators

The goal of this paper is to present an approach to Hod Pair Capturing (HPC). $HPC$ is the most outstanding open problem of descriptive inner model theory. More specifically, we introduce two principles, the Direct Limit Independence and the Bounded Direct Limits, and show that they together imply HPC.

math.LO↗

arXiv2025

AD$^+$ implies that $ω_1$ is a $Θ$-Berkeley cardinal

Following \cite{bagaria2019large}, given cardinals $κ<λ$, we say $κ$ is a club $λ$-Berkeley cardinal if for every transitive set $N$ of size $<λ$ such that $κ\subseteq N$, there is a club $C\subseteq κ$ with the property that for every $η\in C$ there is an elementary embedding $j: N\rightarrow N$ with crit$(j)=η$. We say $κ$ is $ν$-club $λ$-Berkeley if $C\subseteq κ$ as above is a $ν$-club. We say $κ$ is $λ$-Berkeley if $C$ is unbounded in $κ$. We show that under AD$^+$, (1) every regular Suslin cardinal is $ω$-club $Θ$-Berkeley (see \rthm{main theorem}), (2) $ω_1$ is club $Θ$-Berkeley (see \rthm{main theorem lr} and \rthm{main theorem}), and (3) the ${\tildeδ}^1_{2n}$'s are $Θ$-Berkeley -- in particular, $ω_2$ is $Θ$-Berkeley (see \rrem{omega2}). Along the way, we represent regular Suslin cardinals in direct limits as cutpoint cardinals (see \rthm{char extenders}). This topic has been studied in \cite{MPSC} and \cite{jackson2022suslin}, albeit from a different point of view. We also show that, assuming $V=L(\mathbb{R})+{\mathrm{AD}}$, $ω_1$ is not $Θ^+$-Berkeley, so the result stated in the title is optimal (see \rthm{lr optimal} and \rthm{thetareg optimal}).

math.LO↗

arXiv2025

A model of the Axiom of Determinacy in which every set of reals is universally Baire

The consistency of the theory $\mathsf{ZF} + \mathsf{AD}_{\mathbb{R}} + {}$``every set of reals is universally Baire'' is proved relative to $\mathsf{ZFC} + {}$``there is a cardinal that is a limit of Woodin cardinals and of strong cardinals.'' The proof is based on the derived model construction, which was used by Woodin to show that the theory $\mathsf{ZF} + \mathsf{AD}_{\mathbb{R}} + {}$``every set of reals is Suslin'' is consistent relative to $\mathsf{ZFC} + {}$``there is a cardinal $λ$ that is a limit of Woodin cardinals and of $\mathord{<}λ$-strong cardinals.'' The $Σ^2_1$ reflection property of our model is proved using genericity iterations as used by Neeman and Steel.

math.LO↗

arXiv2025

Chang models over derived models with supercompact measures

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 $\mathsf{AD}_{\mathbb{R}} + Θ$ is regular + $ω_1$ is ${<}δ_{\infty}$-supercompact for some regular cardinal $δ_{\infty}>Θ$. This complements Woodin's generalized Chang model, which satisfies $\mathsf{AD}_{\mathbb{R}}+ω_1$ is supercompact, assuming a proper class of Woodin cardinals that are limits of Woodin cardinals.

math.LO↗

arXiv2025

The Largest Suslin Axiom

We develop the basic fine structure theory of the minimal model of the Largest Suslin Axiom. In particular, we prove that that the minimal model of the Largest Suslin Axiom satisfies the Mouse Set Conjecture, and that the Proper Forcing Axiom implies the minimal model of the Largest Suslin Axiom exists.

math.LO↗

arXiv2025

Nairian Models

We introduce a hierarchy of models of the Axiom of Determinacy called \emph{Nairian models}. Forcing over the simplest Nairian model, we obtain a model of ${\sf{ZFC}}+{\sf{MM^{++}}}(c)+\neg\square_{ω_3}+\neg\square(ω_3)$. Then, fixing $n\in [3, ω)$, we design a Nairian model and force over it to produce a model of ${\sf{ZFC}}+{\sf{MM^{++}}}(c)+\forall i\in [2, n]\, \neg\square(ω_i)$. We also build a Nairian model that satisfies ${\sf{ZF}}+"ω_1$ is a supercompact cardinal." We obtain as corollaries of these constructions (1) the consistent failure of the Iterability Conjecture for the Mitchell-Schindler $\sf{K}^{c}$ construction, (2) the consistent failure of the Iterability Conjecture for the $\sf{K}^{c}$ construction using $2^{2^{\dots 2^ω}}$-complete (for any finite stack of exponents) background extenders, answering a strong version of a question asked by Steel, and (3) a negative answer to Trang's question whether ${\sf{ZF}}+"ω_1$ is a supercompact cardinal" is equiconsistent with ${\sf{ZFC}}+"$there is a proper class of Woodin cardinals that are limits of Woodin cardinals." These corollaries identify obstructions to extending the methods of (descriptive) inner model theory past a Woodin cardinal which is a limit of Woodin cardinals.

math.LO↗

arXiv2025

Building Models of Determinacy from Below

We present an $L$-like construction that produces the minimal model of $\mathsf{AD}_\mathbb{R}+$"$Θ$ is regular". In fact, our construction can produce any model of $\mathsf{AD}^++\mathsf{AD}_\mathbb{R}+V=L(P(\mathbb{R}))$ in which there is no hod mouse with a measurable limit of Woodins.

math.LO↗

arXiv2024

Gödel's Program in Set Theory

Gödel proved in the 1930s in his famous Incompleteness Theorems that not all statements in mathematics can be proven or disproven from the accepted ZFC axioms. A few years later he showed the celebrated result that Cantor's Continuum Hypothesis is consistent. Afterwards, Gödel raised the question whether, despite the fact that there is no reasonable axiomatic framework for all mathematical statements, natural statements, such as Cantor's Continuum Hypothesis, can be decided via extending ZFC by large cardinal axioms. While this question has been answered negatively, the problem of finding good axioms that decide natural mathematical statements remains open. There is a compelling candidate for an axiom that could solve Gödel's problem: V = Ultimate-L. In addition, due to recent results the Sealing scenario has gained a lot of attention. We describe these candidates as well as their impact and relationship.

math.LO↗