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Sean Cox

Publications and source records attributed to Sean Cox.

31 records · Page 2Linked to original sources
arXiv2020

The $Π^1_1 \! \! \downarrow$ Löwenheim-Skolem-Tarski property of Stationary Logic

Fuchino-Maschio-Sakai~\cite{FuchinoEtAl_DRP_LST} proved that the Löwenheim-Skolem-Tarski (LST) property of Stationary Logic is equivalent to the Diagonal Reflection Principle on internally club sets ($\text{DRP}_{\text{IC}}$) introduced in \cite{DRP}. We prove that the restriction of the LST property to (downward) reflection of $Π^1_1$ formulas, which we call the $Π^1_1 \! \! \downarrow$-LST property, is equivalent to the \emph{internal} version of DRP from \cite{Cox_RP_IS}. Combined with results from \cite{Cox_RP_IS}, this shows that the $Π^1_1 \! \! \downarrow$-LST Property for Stationary Logic is strictly weaker than the full LST Property for Stationary Logic, though if CH holds they are equivalent.

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arXiv2019

Club Chang's Conjecture

Chang's Conjecture (CC) asserts that for every $F:[ω_2]^{<ω} \to ω_2$, there exists an $X$ that is closed under $F$ such that $|X|=ω_1$ and $|X \cap ω_1| =ω$. By classic results of Silver and Donder, CC is equiconsistent with an $ω_1$-Erdos cardinal. Using stronger large cardinal assumptions (between $o(κ) = κ^+$ and $o(κ) = κ^{++}$), we prove that it is consistent to also require that $X$ contains a closed unbounded set of ordinals in $\text{sup}(X \cap ω_2)$. We denote this stronger principle \textbf{Club-CC}, and also show that, unlike CC, Club-CC implies failure of certain weak square principles.

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arXiv2019

Indestructible Guessing Models and the Continuum

We introduce a stronger version of an $ω_1$-guessing model, which we call an indestructibly $ω_1$-guessing model. The principle IGMP states that there are stationarily many indestructibly $ω_1$-guessing models. This principle, which follows from PFA, captures many of the consequences of PFA, including the Suslin hypothesis and the singular cardinal hypothesis. We prove that IGMP is consistent with the continuum being arbitrarily large.

math.LO↗
arXiv2018

A variant of Shelah's characterization of Strong Chang's Conjecture

Shelah considered a certain version of Strong Chang's Conjecture, which we denote $\text{SCC}^{\text{cof}}$, and proved that it is equivalent to several statements, including the assertion that Namba forcing is semiproper. We introduce an apparently weaker version, denoted $\text{SCC}^{\text{split}}$, and prove an analogous characterization of it. In particular, $\text{SCC}^{\text{split}}$ is equivalent to the assertion that the the Friedman-Krueger poset is semiproper. This strengthens and sharpens the results of Cox, and sheds some light on problems from Usuba and Torres-Perez and Wu.

math.LO↗
arXiv2018

Namba forcing, weak approximation, and guessing

We prove a variation of Easton's lemma for strongly proper forcings, and use it to prove that, unlike the stronger principle $\textsf{IGMP}$, $\textsf{GMP}$ together with $2^ω\le ω_2$ is consistent with the existence of an $ω_1$-distributive nowhere c.c.c. forcing poset of size $ω_1$. We introduce the idea of a weakly guessing model, and prove that many of the strong consequences of the principle $\textsf{GMP}$ follow from the existence of stationarily many weakly guessing models. Using Namba forcing, we construct a model in which there are stationarily many indestructibly weakly guessing models which have a bounded countable subset not covered by any countable set in the model.

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arXiv2018

Strongly proper forcing and some problems of Foreman

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}.

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arXiv2017

On the universality of the nonstationary ideal

Burke \cite{MR1472122} proved that the generalized nonstationary ideal, denoted NS, is universal in the following sense: every normal ideal, and every tower of normal ideals of inaccessible height, is a canonical Rudin-Keisler projection of the restriction of $\text{NS}$ to some stationary set. We investigate how far Burke's theorem can be pushed, by analyzing the universality properties of NS with respect to the wider class of \emph{$\mathcal{C}$-systems of filters} introduced by Audrito-Steila \cite{AudritoSteila}. First we answer a question of \cite{AudritoSteila}, by proving that $\mathcal{C}$-systems of filters do not capture all kinds of set-generic embeddings. We provide a characterization of supercompactness in terms of short extenders and canonical projections of NS, without any reference to the strength of the extenders; as a corollary, NS can consistently fail to canonically project to arbitrarily strong short extenders. We prove that $ω$-cofinal towers of normal ultrafilters---e.g.\ the kind used to characterize I2 and I3 embeddings---are well-founded if and only if they are canonical projections of NS. Finally, we provide a characterization of "$\aleph_ω$ is Jonsson" in terms of canonical projections of NS.

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arXiv2016

Characterizing large cardinals in terms of layered posets

Given an uncountable regular cardinal $κ$, a partial order is $κ$-stationarily layered if the collection of regular suborders of $\mathbb{P}$ of cardinality less than $κ$ is stationary in $\mathcal{P}_κ(\mathbb{P})$. We show that weak compactness can be characterized by this property of partial orders by proving that an uncountable regular cardinal $κ$ is weakly compact if and only if every partial order satisfying the $κ$-chain condition is $κ$-stationarily layered. We prove a similar result for strongly inaccessible cardinals. Moreover, we show that the statement that all $κ$-Knaster partial orders are $κ$-stationarily layered implies that $κ$ is a Mahlo cardinal and every stationary subset of $κ$ reflects. This shows that this statement characterizes weak compactness in canonical inner models. In contrast, we show that it is also consistent that this statement holds at a non-weakly compact cardinal.

math.LO↗
arXiv2015

Indestructibility of generically strong cardinals

Foreman proved the Duality Theorem, which gives an algebraic characterization of certain ideal quotients in generic extensions. As an application he proved that generic supercompactness of $ω_1$ is preserved by any proper forcing. We generalize portions of Foreman's Duality Theorem to the context of generic extender embeddings and ideal extenders (as introduced by Claverie in his PhD Thesis, Universitat Munster, 2010). As an application we prove that if $ω_1$ is generically strong, then it remains so after adding any number of Cohen subsets of $ω_1$; however many other $ω_1$-closed posets---such as $\text{Col}(ω_1, ω_2)$---can destroy the generic strength of $ω_1$. This generalizes some results of Gitik-Shelah about indestructibility of strong cardinals to the generically strong context. We also prove similar theorems for successor cardinals larger than $ω_1$.

math.LO↗
arXiv2015

Quotients of Strongly Proper Forcings and Guessing Models

We prove that a wide class of strongly proper forcing posets have quotients with strong properties. Specifically, we prove that quotients of forcing posets which have simple universal strongly generic conditions on a stationary set of models by certain nice regular suborders satisfy the $ω_1$-approximation property. We prove that the existence of stationarily many $ω_1$-guessing models in $P_{ω_2}(H(θ))$, for sufficiently large cardinals $θ$, is consistent with the continuum being arbitrarily large, solving a problem of Viale and Weiss.

math.LO↗
arXiv2011

Martin's Maximum and tower forcing

There are several examples in the literature showing that compactness-like properties of a cardinal $κ$ cause poor behavior of some generic ultrapowers which have critical point $κ$ (Burke \cite{MR1472122} when $κ$ is a supercompact cardinal; Foreman-Magidor \cite{MR1359154} when $κ= ω_2$ in the presence of strong forcing axioms). We prove more instances of this phenomenon. First, the Reflection Principle (RP) implies that if $\vec{\mathcal{I}}$ is a tower of ideals which concentrates on the class $GIC_{ω_1}$ of $ω_1$-guessing, internally club sets, then $\vec{\mathcal{I}}$ is not presaturated (a set is $ω_1$-guessing iff its transitive collapse has the $ω_1$-approximation property as defined in Hamkins \cite{MR2540935}). This theorem, combined with work from \cite{VW_ISP}, shows that if $PFA^+$ or $MM$ holds and there is an inaccessible cardinal, then there is a tower with critical point $ω_2$ which is not presaturated; moreover this tower is significantly different from the non-presaturated tower already known (by Foreman-Magidor \cite{MR1359154}) to exist in all models of Martin's Maximum. The conjunction of the Strong Reflection Principle (SRP) and the Tree Property at $ω_2$ has similar implications for towers of ideals which concentrate on the wider class $GIS_{ω_1}$ of $ω_1$-guessing, internally stationary sets. Finally, we show that the word "presaturated" cannot be replaced by "precipitous" in the theorems above: Martin's Maximum (which implies SRP and the Tree Property at $ω_2$) is consistent with a precipitous tower on $GIC_{ω_1}$.

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