arXiv subjects
Publications and source records attributed to Grigor Sargsyan.
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.
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.
This white paper is the result of a collaboration by those that attended a workshop at the Facility for Rare Isotope Beams (FRIB), organized by the FRIB Theory Alliance (FRIB-TA), on Theoretical Justifications and Motivations for Early High-Profile FRIB Experiments. It covers a wide range of topics related to the science that will be explored at FRIB. After a brief introduction, the sections address: (II) Overview of theoretical methods, (III) Experimental capabilities, (IV) Structure, (V) Near-threshold Physics, (VI) Reaction mechanisms, (VII) Nuclear equations of state, (VIII) Nuclear astrophysics, (IX) Fundamental symmetries, and (X) Experimental design and uncertainty quantification.
In the context of $\mathsf{ZF}+\mathsf{DC}$, we force $\mathsf{DC}_κ$ for relations on $\mathcal{P}(κ)$ for $κ<\aleph_ω$ over the Chang model $\mathrm{L}(\mathrm{Ord}^ω)$ making some assumptions on the thorn sequence defined by ${\it \unicode{xFE}}_0=ω$, ${\it \unicode{xFE}}_{α+1}$ as the least ordinal not a surjective image of ${\it \unicode{xFE}}_α^ω$ (i.e. no $f:{\it \unicode{xFE}}_α^ω\rightarrow {\it \unicode{xFE}}_{α+1}$ is surjective) and ${\it \unicode{xFE}}_γ=\sup_{α<γ}{\it \unicode{xFE}}_α$ for limit $γ$. These assumptions are motivated from results about $Θ$ in the context of determinacy, and could be reasonable ways of thinking about the Chang model. Explicitly, we assume cardinals $λ$ on the thorn sequence are strongly regular (meaning regular and functions $f:κ^{<κ}\rightarrow λ$ are bounded whenever $κ<λ$ is on the thorn sequence) and justified (meaning $\mathcal{P}(κ^ω)\cap \mathrm{L}(\mathrm{Ord}^ω)\subseteq \mathrm{L}_λ(λ^ω,X)$ for some $X\subseteq λ$ for any $κ<λ$ on the thorn sequence). This allow us to use Cohen forcing and establish more dependent choice.
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 $\mathsf{AD}^+$ in any set generic extensions.
We show that in extender models there are no generic embeddings with critical point $ω_1$ that resemble the stationary tower at the second Woodin cardinal.
We show that the following two theories are equiconsistent: (T) ZFC, CH and "There is a dense ideal on the first uncountable cardinal such that if j is the generic embedding associated with it then its restriction on ordinals is independent of the generic object is". (S) ZF, ADR and "Theta is a regular cardinal." The main result of this paper is that T implies that the minimal model of S exists. Woodin, in unpublished work, showed that the consistency of S implies the consistency of T. We will also give a proof of this result, which, together with our main theorem, establishes the equiconsistency of T and S. Our main result partially resolves a well-known conjecture of Woodin, and completely solves one of the main Core Model Induction problems dating back to 90s.
We obtain a partial result on the following conjecture. Conjecture. Let (P, Σ) be a projectum stable mouse pair, and let κbe a cardinal of V such that κ< o(M_\infty(P, Σ)); then the following are equivalent: (1) κis a Suslin cardinal, (2) κis a cutpoint of M_\infty(P, Σ).
Suppose $M$ is a transitive class size model of $AD_{\mathbb{R}}+``Θ$ is regular". $M$ is a minimal model of $AD_{\mathbb{R}}+``Θ$ is measurable" if (i) $\mathbb{R}, Ord\subseteq M$ (ii) there is $μ\in M$ such that $M\models ``μ$ is a normal $\mathbb{R}$-complete measure on $Θ$" and (iii) for any transitive class size $N\subsetneq M$ such that $\mathbb{R}\subseteq N$, $N\models ``$there is no $\mathbb{R}$-complete measure on $Θ$". Continuing Trang's work in [8], we compute HOD of a minimal model of $AD_{\mathbb{R}}+``Θ$ is measurable".
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 (see for example [6]). However, the proof uses a different set of ideas and is more general.
We present a covering conjecture that we expect to be true below superstrong cardinals. We then show that the conjecture is true in hod mice. This work is a continuation of the work that started in Covering with Universally Baire Functions Advances in Mathematics, and the main conjecture of the current paper is a revision of the UB Covering Conjecture of the aforementioned paper.
$AD_{\mathbb{R}}$ is the Axiom of Determinacy for games on the reals (i.e., the player play reals. We show that it implies that all sets of reals are Theta-universally Baire. As a corollary we obtain a model in which $AD$ holds and all sets of reals are fully universally Baire. Such a model was first constructed by Larson, Sargsyan and Wilson. Our result complements it as the technique is purely descriptive set theoretic.
Recall that the Mouse Set Conjecture says that under AD++V=L(P(R)), a real is ordinal definable if and only if it belongs to an iterable mouse. The Mouse Set Conjecture for sets of reals says that under the same theory, a set of reals is ordinal definable from a real if and only if it belongs to a mouse over the reals. We prove that the Mouse Set Conjecture implies the Mouse Set Conjecture for sets of reals.
We develop a basic translation procedure that translates a hod mouse to an equivalent mouse. Unlike the translation procedure used by Steel and Zhu, our procedure works in a coarse setting without assuming AD. Nevertheless, the procedure resembles the one developed by Steel. We use the translation procedures to answer a question of Trevor Wilson. Namely, we show that if there is a stationary class of lambda such that lambda is a limit of Woodin cardinals and the derived model at lambda satisfies AD the first member of the Solovay sequence is is less than Theta then there is a transitive model M such that M contains the ordinals and satisfies there is a proper class of Woodin cardinals and a strong cardinal.
We obtain sealing by forcing over a self-iterable model. The proof is fine-structure free and uses only basic ideas from iteration theory. We believe that such fine-structure free proofs will make the subject more accessible to the general set theoretic community.
We give a new proof of a theorem of Becker that under AD+V=L(R), omega_2 is a kappa-supercompact for every kappa less than or equal to the supremum of all Suslin cardinals. Our proof uses inner model theory. It is still open whether one can prove, say under AD_R, that Suslin cardinals and their successors are <Theta-supercompact. That this is true is a theorem of Steve Jackson, who proved it using generic codes.
Assume $AD+V=L(\mathbb{R})$. Let $κ=\utildeδ^2_1$, the supremum of all $\utildeΔ^2_1$ prewellorderings. We prove that extenders on the sequence of $\H$ that have critical point $κ$ are generated by countably complete measures. This provides a partial reversal of Woodin's result that the $<Θ$-strongness of $κ$ in $\H$ is witnessed by $κ$-complete ultrafilters on $\k$. The aforementioned characterization of extenders works in a more general setting for all cutpoint measurable cardinals of $\H$ in all models of determinacy where the fine structural analysis of $\H$ has been carried out. For example, it holds in the minimal model of the Largest Suslin Axiom. It also gives a simple proof of a theorem of Steel that the successor members of the Solovay sequence are cutpoints in $\H$ (in models where $\H$ analysis is carried out).
We show that the statements $\square(ω_{3})$ and $\square(ω_{4})$ both fail in the $\mathbb{P}_{\mathrm{max}}$ extension of a variation of the Chang model introduced by Sargsyan.