arXiv subjects
Publications and source records attributed to Yair Hayut.
We study which $κ$-distributive forcing notions of size $κ$ can be embedded into tree Prikry forcing notions with $κ$-complete ultrafilters under various large cardinal assumptions. An alternative formulation -- can the filter of dense open subsets of a $κ$-distributive forcing notion of size $κ$ be extended to a $κ$-complete ultrafilter.
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 cardinal $κ$ the nonexistence of a tree on $κ$ with exactly $κ^{+}$ many branches and, in particular, the Perfect Subtree Property for $κ$, implies the consistency of $AD_{\mathbb{R}} + DC$.
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 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 $\aleph_{ω^ω}$.
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 $\aleph_{ω^2}$.
We show that if $λ^{<κ} = λ$ and every normal filter on $P_κλ$ can be extended to a $κ$-complete ultrafilter then so does every $κ$-complete filter on $λ$. This answers a question of Gitik.
We prove that the consistency of the existence of a Dowker filter at $κ^+$ along with $2^κ=κ^+$ where $κ$ is regular and uncountable. Using Magidor forcing we also prove the consistency of the existence of a Dowker filter at $μ^+$ where $μ>{\rm cf}(μ)>ω$.
We show that for many pairs of infinite cardinals $κ> μ^+ > μ$, $(κ^{+}, κ)\twoheadrightarrow (μ^+, μ)$ is consistent relative to the consistency of a supercompact cardinal. We also show that it is consistent, relative to a huge cardinal that $(κ^{+}, κ)\twoheadrightarrow (μ^+, μ)$ for every successor cardinal $κ$ and every $μ< κ$, answering a question of Foreman.
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.
In this paper we study the notion of $C^{(n)}$-supercompactness introduced by Bagaria in \cite{Bag} 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 $C^{(n)}$-supercompact). Furthermore, we prove under suitable hypothesis that the ultimate identity crises is also possible. These results solve several questions posed by Bagaria and Tsaprounis.
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 prove a technical criterion for lifting elementary embeddings to symmetric extensions, and we use this to show that it is consistent relative to a supercompact cardinal that there is a critical cardinal whose successor is singular.
We prove that $α_M(λ)$ can be successor of a supercompact cardinal, when $λ$ is a Magidor cardinal. From this result we obtain the consistency of $α_M(λ)$ being a successor of a singular cardinal with uncountable cofinality.
We construct a model in which the tree property holds in $\aleph_{ω+ 1}$ and it is destructible under $\text{Col}(ω, ω_1)$. On the other hand we discuss some cases in which the tree property is indestructible under small or closed forcings.
In this paper we analyze the connection between some properties of partially strongly compact cardinals: the completion of filters of certain size and instances of the compactness of $\mathcal{L}_{κ,κ}$. Using this equivalence we show that if any $κ$-complete filter on $λ$ can be extended to a $κ$-complete ultrafilter and $λ^{<κ} = λ$ then $\square(μ)$ fails for all regular $μ\in[κ,2^λ]$. As an application, we improve the lower bound for the consistency strength of $κ$-compactness, a case which was explicitly considered by Mitchell.
In this paper we investigate the consequences and consistency of the downward Löwenheim-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 successor of singular cardinals.
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 successors of regular cardinals which carry uniform ultrafilters; we also show that this spectrum is not necessarily closed.
We analyze the non-stationary ideal and the club filter at aleph_1 under MM.
We improve the upper bound for the consistency strength of stationary reflection at successors of singular cardinals.