Yair Hayut

Hebrew University of Jerusalem | HUJI

In this paper, we obtain the consistency, relative to large cardinals, of the existence of dense ideals on every successor of a regular cardinal simultaneously. Using a consequent transfer principle, we show that in this model there is a $\sigma$-complete, $\aleph_1$-dense ideal on $\aleph_{n+1}$ for every $n < \omega$, answering a question of Fore...

We introduce a new compactness principle which we call the gluing property. For a measurable cardinal [Formula: see text] and a cardinal [Formula: see text], we say that [Formula: see text] has the [Formula: see text]-gluing property if every sequence of [Formula: see text]-many [Formula: see text]-complete ultrafilters on [Formula: see text] can b...

The following question was asked by Grigorieff: Suppose $V$ is a ZFC model and $V[G]$ is a set-generic extension of $V$. Can there be a ZF model $N$ so that $V\subset N \subset V[G]$ yet $N$ is not equal to $V(A)$ for any set $A\in V[G]$? The first such model was constructed by Karagila. This is the so-called \emph{Bristol model}, an intermediate m...

In this paper we prove that from large cardinals it is consistent that there is a singular strong limit cardinal $\nu $ such that the singular cardinal hypothesis fails at $\nu $ and every collection of fewer than $\operatorname {\mathrm {cf}}(\nu )$ stationary subsets of $\nu ^{+}$ reflects simultaneously. For $\operatorname {\mathrm {cf}}(\nu )>...

We study which [Formula: see text]-distributive forcing notions of size [Formula: see text] can be embedded into tree Prikry forcing notions with [Formula: see text]-complete ultrafilters under various large cardinal assumptions. An alternative formulation — can the filter of dense open subsets of a [Formula: see text]-distributive forcing notion o...

This paper is meant to present in a coherent way several instances of quite common phenomena that was first identified (independently) by Bukovsk\'y and Dehornoy. We present the basic result for Prikry type forcing and show how to extend it to the Gitik-Shraon forcing, the Extender Based Prikry forcing, Prikry forcings with interleaved collapses an...

In this paper we study the spectrum of heights of transitive models of theories extending $V = L[A]$, under various definitions. In particular, we investigate the consistency strength of making those spectra as simple as possible.

We prove several consistency results concerning the notion of $\omega $ -strongly measurable cardinal in $\operatorname {\mathrm {HOD}}$ . In particular, we show that is it consistent, relative to a large cardinal hypothesis weaker than $o(\kappa ) = \kappa $ , that every successor of a regular cardinal is $\omega $ -strongly measurable in $\operat...

We continue to work from [5] and make a small -- but significant -- improvement to the definition of $j$-decomposable system. This provides us with a better lifting of elementary embeddings to symmetric extension. In particular, this allows us to more easily lift weakly compact embedding and thus preserve the notion of weakly critical cardinals. We...

We introduce a new compactness principle which we call the gluing property. For a measurable cardinal $\kappa$ and a cardinal $\lambda$, we say that $\kappa$ has the $\lambda$-gluing property if every sequence of $\lambda$-many $\kappa$-complete ultrafilters on $\kappa$ can be glued into a $\kappa$-complete extender. We show that every $\kappa$-com...

We investigate the consistency strength of the statement: κ is weakly compact and there is no tree on κ with exactly κ+ many branches. We show that this statement fails strongly (in the sense that there is a sealed tree with exactly κ+ many branches) if there is no inner model with a Woodin cardinal. Moreover, we show that for a weakly compact card...

We provide a model theoretical and tree property-like characterization of $\lambda $ - $\Pi ^1_1$ -subcompactness and supercompactness. We explore the behavior of these combinatorial principles at accessible cardinals.

We study which $\kappa$-distributive forcing notions of size $\kappa$ can be embedded into tree Prikry forcing notions with $\kappa$-complete ultrafilters under various large cardinal assumptions. An alternative formulation -- can the filter of dense open subsets of a $\kappa$-distributive forcing notion of size $\kappa$ be extended to a $\kappa$-c...

We introduce invariants for compact $C^1$-orientable surfaces (with boundary) in $\mathbb{R}^3$. Our invariants are certain degree four polynomials in the moments of the delta function of the surface. We give an effective and numerically stable inversion algorithm for getting the surface from the invariants, which works on a co-meager subset of $C^...

We investigate systems of transitive models of ZFC which are elementarily embeddable into each other and the influence of definability properties on such systems.

We study the Galvin property. We show that various square principles imply that the cofinality of the Galvin number is uncountable (or even greater than \(\aleph _1\)). We prove that the proper forcing axiom is consistent with a strong negation of the Glavin property.

We investigate the possibilities of global versions of Chang’s Conjecture that involve singular cardinals. We show some ZFC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document...

We improve the upper bound for the consistency strength of stationary reflection at successors of singular cardinals.

In this paper we study the notion of $C^{(n)}$ -supercompactness introduced by Bagaria in [3] and prove the identity crises phenomenon for such class. Specifically, we show that consistently the least supercompact is strictly below the least $C^{(1)}$ -supercompact but also that the least supercompact is $C^{(1)}$ -supercompact (and even $C^{(n)}$...

THE TREE PROPERTY AT THE TWO IMMEDIATE SUCCESSORS OF A SINGULAR CARDINAL - JAMES CUMMINGS, YAIR HAYUT, MENACHEM MAGIDOR, ITAY NEEMAN, DIMA SINAPOVA, SPENCER UNGER

We prove that the existence of a Dowker filter at κ + \kappa ^+ , where κ \kappa is regular and uncountable, is consistent with 2 κ = κ + 2^\kappa =\kappa ^+ . We also prove the consistency of a Dowker filter at μ + \mu ^+ where μ > cf ( μ ) > ω \mu >\text {cf}(\mu )>\omega . This can be forced with 2 μ = μ + 2^\mu =\mu ^+ as well.

We correct a flawed proof from our paper, "On the consistency of local and global versions of Chang’s conjecture."

We introduce the notion of a critical cardinal as the critical point of sufficiently strong elementary embedding between transitive sets. Assuming the Axiom of Choice this is equivalent to measurability, but it is wellknown that Choice is necessary for the equivalence. Oddly enough, this central notion was never investigated on its own before. We p...

ITP is a combinatorial principle that is a strengthening of the tree property. For an inaccessible cardinal κ \kappa , ITP at κ \kappa holds if and only if κ \kappa is supercompact. And just like the tree property, it can be forced to hold at accessible cardinals. A broad project is obtaining ITP at many cardinals simultaneously. Past a singular ca...

We provide a model theoretical and tree property like characterization of $\lambda$-$\Pi^1_1$-subcompactness and supercompactness. We explore the behaviour of those combinatorial principles at accessible cardinals.

We show that if λ > κ = λ \lambda ^{>\kappa } = \lambda and every normal filter on P κ λ P_\kappa \lambda can be extended to a κ \kappa -complete ultrafilter, then so does every κ \kappa -complete filter on λ \lambda . This answers a question of Gitik.

We prove several consistency results concerning the notion of $\omega$-strongly measurable cardinal in HOD. In particular, we show that is it consistent, relative to a large cardinal hypothesis weaker than $o(\kappa) = \kappa$, that every successor of a regular cardinal is $\omega$-strongly measurable in HOD.

We investigate the consistency strength of the statement: $\kappa$ is weakly compact and there is no tree on $\kappa$ with exactly $\kappa^{+}$ many branches. We show that this statement fails strongly (in the sense that there is a sealed tree with exactly $\kappa^{+}$ many branches) if there is no inner model with a Woodin cardinal. Moreover, we s...

We prove that the consistency of the existence of a Dowker filter at $\kappa^+$ along with $2^\kappa=\kappa^+$ where $\kappa$ is regular and uncountable. Using Magidor forcing we also prove the consistency of the existence of a Dowker filter at $\mu^+$ where $\mu>{\rm cf}(\mu)>\omega$.

We show that if $\lambda^{<\kappa} = \lambda$ and every normal filter on $P_\kappa\lambda$ can be extended to a $\kappa$-complete ultrafilter then so does every $\kappa$-complete filter. This answers a question of Gitik.

In this paper we prove that from large cardinals it is consistent that there is a singular strong limit cardinal $\nu$ such that the singular cardinal hypothesis fails at $\nu$ and every collection of fewer than $\mathrm{cf}(\nu)$ stationary subsets of $\nu^+$ reflects simultaneously. For uncountable cofinality, this situation was not previously kn...

DESTRUCTIBILITY OF THE TREE PROPERTY AT אω+1 - YAIR HAYUT, MENACHEM MAGIDOR

In this paper we study the notion of C (n)-supercompactness introduced by Bagaria in [Bag12] and prove the identity crises phenomenon for such classes. Specifically , we show that consistently the least supercompact is strictly below the least C (1)-supercompact but also that the least (ω 1-)strong compact is C (1)-supercompact (and even C (n)-supe...

We investigate the possibilities of global versions of Chang's Conjecture that involve singular cardinals. We show some $\mathrm{ZFC}$ limitations on such principles, and prove relative to large cardinals that Chang's Conjecture can consistently hold between all pairs of limit cardinals below $\aleph_{\omega^\omega}$.

We introduce the notion of a critical cardinal as the critical point of sufficiently strong elementary embedding between transitive sets. Assuming the axiom of choice this is equivalent to measurability, but it is well-known that choice is necessary for the equivalence. Oddly enough, this central notion was never investigated on its own before. We...

We improve the upper bound for the consistency strength of stationary reflection at successors of singular cardinals.

In this paper we analyze the connection between some properties of partially strongly compact cardinals: the completion of filters and the compactness of $\mathcal{L}_{\kappa,\kappa}$. We show that if any $\kappa$-complete filter on $\lambda$ can be extended to a $\kappa$-complete ultrafilter and $\lambda^{<\kappa} = \lambda$ then $\square(\mu)$ fa...

We prove that αM (λ) can be successor of a supercompact cardinal, when λ is a Magidor cardinal. From this result we obtain the consistency of αM (λ) being a successor of a singular cardinal with uncountable cofinality. 2010 Mathematics Subject Classification. 03E55.

We study some limitations and possible occurrences of uniform ultrafilters on ordinals without the axiom of choice. We prove an Easton-like theorem about the possible spectrum of regular cardinals which carry uniform ultrafilters; we also show that this spectrum is not necessarily closed.

In this paper we investigate the consequences and consistency of the downward L\"{o}wenheim-Skolem theorem for extension of the first order logic by the Magidor-Malitz quantifier. We derive some combinatorial results and improve the known upper bound for the consistency of Chang's conjecture at successors of singular cardinals.

We prove the consistency of the failure of the weak diamond $\Phi_\lambda$ at strongly inaccessible cardinals. On the other hand we show that the very weak diamond $\Psi_\lambda$ is equivalent to the statement $2^{<\lambda}<2^\lambda$ and hence holds at every strongly inaccessible cardinal.

Assuming the existence of a proper class of supercompact cardinals, we force that for every regular cardinal κ, there are κ +-Aronszajn trees and all such trees are special.

We show that for many pairs of cardinals infinite $\kappa > \mu^+ > \mu$, $(\kappa^{+}, \kappa)\twoheadrightarrow (\mu^+, \mu)$ is consistent relative to the consistency of a supercompact cardinal. We also show that it is consistent, relative to a huge cardinal that $(\kappa^{+}, \kappa)\twoheadrightarrow (\mu^+, \mu)$ for every successor cardinal...

Starting from a stationary set of supercompact cardinals we find a generic extension in which the tree property holds at every regular cardinal between $\aleph_2$ and $\aleph_{\omega^2}$.

In this paper we investigate some properties of forcing which can be considered "nice" in the context of singularizing regular cardinals to have an uncountable cofinality. We show that such forcing which changes cofinality of a regular cardinal, cannot be too nice and must cause some "damage" to the structure of cardinals and stationary sets.

Starting from infinitely many supercompact cardinals, we force a model of ZFC where $\aleph_{\omega^2+1}$ satisfies simultaneously a strong principle of reflection, called $\Delta$-reflection, and a version of the square principle, denoted $\square(\aleph_{\omega^2+1}).$ Thus we show that $\aleph_{\omega^2+1}$ can satisfy simultaneously a strong re...

We construct a model in which the tree property holds in $\aleph_{\omega + 1}$ and it is destructible under $\text{Col}(\omega, \omega_1)$. On the other hand we discuss some cases in which the tree property is indestructible under small or closed forcings.

We investigate the relationship between weak square principles and simultaneous reflection of stationary sets.

We show that it is consistent, relative to $\omega$ many supercompact cardinals, that the super tree property holds at $\aleph_n$ for all $2 \leq n < \omega$ but there are weak square and a very good scale at $\aleph_{\omega}$.

We introduce a model-theoretic characterization of Magidor cardinals, from which we infer that Magidor filter are beyond ZFC-inconsistency

Starting from the existence of many supercompact cardinals, we construct a model of $ZFC+GCH$ in which the tree property holds at a countable segment of successor of singular cardinals.

We define Magidor cardinals as Jonsson cardinals, but the coloring is defined on $\omega$-bounded subsets. We prove that Magidor cardinals appear in a fairly high level of the chart of large cardinals. In particular, a limit of I1 cardinals need not be a Magidor cardinal.

In this paper we consider the Foreman's maximality principle, which says that any non-trivial forcing notion either adds a new real or collapses some cardinals. We prove the consistency of some of its consequences. We prove that it is consistent that every $c.c.c.$ forcing adds a real and that for every uncountable regular cardinal $\kappa$, every...