arXiv subjects
Publications and source records attributed to Monroe Eskew.
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.
An ideal $I$ on a cardinal $κ$ is called \emph{rigid} if all automorphisms of $P(κ)/I$ are trivial. An ideal is called \emph{$μ$-minimal} if whenever $G\subseteq P(κ)/I$ is generic and $X\in P(μ)^{V[G]}\setminus V$, it follows that $V[X]=V[G]$. We prove that the existence of a rigid saturated $μ$-minimal ideal on $μ^+$, where $μ$ is a regular cardinal, is consistent relative to the existence of large cardinals. The existence of such an ideal implies that GCH fails. However, we show that the existence of a rigid saturated ideal on $μ^+$, where $μ$ is an \emph{uncountable} regular cardinal, is consistent with GCH relative to the existence of an almost-huge cardinal. Addressing the case $μ=ω$, we show that the existence of a rigid \emph{presaturated} ideal on $ω_1$ is consistent with CH relative to the existence of an almost-huge cardinal. The existence of a \emph{precipitous} rigid ideal on $μ^+$ where $μ$ is an uncountable regular cardinal is equiconsistent with the existence of a measurable cardinal.
A forest is a generalization of a tree, and here we consider the Aronszajn and Suslin properties for forests. We focus on those forests satisfying coherence, a local smallness property. We show that coherent Aronszajn forests can be constructed within ZFC. We give several ways of obtaining coherent Suslin forests by forcing, one of which generalizes the well-known argument of Todorčević that a Cohen real adds a Suslin tree. Another uses a strong combinatorial principle that plays a similar role to diamond. We show that, starting from a large cardinal, this principle can be obtained by a forcing that is small relative to the forest it constructs.
Improving a result of Woodin, we identify some classes of individually consistent but mutually inconsistent generic large cardinal axioms.
We extend prior results of Cody-Eskew, showing the consistency of GCH with the statement that for all regular cardinals $κ\leq λ$, where $κ$ is the successor of a regular cardinal, there is a rigid saturated ideal on $\mathcal{P}_κλ$. We also show the consistency of some instances of rigid saturated ideals on $\mathcal{P}_κλ$ where $κ$ is the successor of a singular cardinal.
We argue against Foreman's proposal to settle the continuum hypothesis and other classical independent questions via the adoption of generic large cardinal axioms.
We provide solutions to several problems of Foreman about ideals, several of which are closely related to Mitchell's notion of \emph{strongly proper} forcing. We prove: 1) Presaturation of a normal ideal implies projective antichain catching, enabling us to provide a solution to a problem from Foreman~\cite{MR2768692} about ideal projections which is more comprehensive and simpler than the solution obtained in \cite{MR3343538}. 2) We solve an older question from Foreman~\cite{MR819932} about the relationship between generic hugeness and generic almost hugeness. 3) Finally, we provide solutions to two technical questions from Foreman~\cite{MR3038554} and \cite{MR2768692} related to his \emph{Duality Theorem}.