arXiv subjects

Sean D. Cox

Publications and source records attributed to Sean D. Cox.

5 recordsLinked to original sources

arXiv2022

Filtration Games and Potentially Projective Modules

The notion of a \textbf{$\boldsymbol{\mathcal{C}}$-filtered} object, where $\mathcal{C}$ is some (typically small) collection of objects in a Grothendieck category, has become ubiquitous since the solution of the Flat Cover Conjecture around the year 2000. We introduce the \textbf{$\boldsymbol{\mathcal{C}}$-Filtration Game of length $\boldsymbol{ω_1}$} on a module, paying particular attention to the case where $\mathcal{C}$ is the collection of all countably presented, projective modules. We prove that Martin's Maximum implies the determinacy of many $\mathcal{C}$-Filtration Games of length $ω_1$, which in turn imply the determinacy of certain Ehrenfeucht-Fraïssé games of length $ω_1$; this allows a significant strengthening of a theorem of Mekler-Shelah-Vaananen \cite{MR1191613}. Also, Martin's Maximum implies that if $R$ is a countable hereditary ring, the class of \textbf{$\boldsymbolσ$-closed potentially projective modules} -- i.e., those modules that are projective in some $σ$-closed forcing extension of the universe -- is closed under $<\aleph_2$-directed limits. We also give an example of a (ZFC-definable) class of abelian groups that, under the ordinary subgroup relation, constitutes an Abstract Elementary Class (AEC) with Löwenheim-Skolem number $\aleph_1$ in some models in set theory, but fails to be an AEC in other models of set theory.

math.LO↗

arXiv2020

Forcing axioms, approachability, and stationary set reflection

We prove a variety of theorems about stationary set reflection and concepts related to internal approachability. We prove that an implication of Fuchino-Usuba relating stationary reflection to a version of Strong Chang's Conjecture cannot be reversed; strengthen and simplify some results of Krueger about forcing axioms and approachability; and prove that some other related results of Krueger are sharp. We also adapt some ideas of Woodin to simplify and unify many arguments in the literature involving preservation of forcing axioms.

math.LO↗

arXiv2018

Chang's Conjecture and semiproperness of nonreasonable posets

Let $\mathbb{Q}$ denote the poset which adds a Cohen real then shoots a club through the complement of $\big( [ω_2]^ω\big)^V$ with countable conditions. We prove that the version of Strong Chang's Conjecture from \cite{MR2965421} implies semiproperness of $\mathbb{Q}$, and that semiproperness of $\mathbb{Q}$---in fact semiproperness of any poset which is sufficiently \emph{nonreasonable} in the sense of Foreman-Magidor~\cite{MR1359154}---implies the version of Strong Chang's Conjecture from \cite{MR2723878} and \cite{MR1261218}. In particular, semiproperness of $\mathbb{Q}$ has large cardinal strength, which answers a question of Friedman-Krueger~\cite{MR2276627}. One corollary of our work is that the version of Strong Chang's Conjecture from \cite{MR2965421} does not imply the existence of a precipitous ideal on $ω_1$.

math.LO↗

arXiv2017

Layered posets and Kunen's universal collapse

We develop the theory of layered posets, and use the notion of layering to prove a new iteration theorem (Theorem 6): if $κ$ is weakly compact then any universal Kunen iteration of $κ$-cc posets (each possibly of size $κ$) is $κ$-cc, as long as direct limits are used sufficiently often. This iteration theorem simplifies and generalizes the various chain condition arguments for universal Kunen iterations in the literature on saturated ideals, especially in situations where finite support iterations are not possible. We also provide two applications: (1) For any $n \ge 1$, a wide variety of $<ω_{n-1}$-closed, $ω_{n+1}$-cc posets of size $ω_{n+1}$ can consistently be absorbed (as regular suborders) by quotients of saturated ideals on $ω_n$ (see Theorem 7 and Corollary 8); and (2) For any $n \in ω$, the Tree Property at $ω_{n+3}$ is consistent with the Chang's Conjecture $(ω_{n+3}, ω_{n+1}) \twoheadrightarrow (ω_{n+1}, ω_n)$ (Theorem 9).

math.LO↗

arXiv2014

Prevalence of Generic Laver Diamond

Viale \cite{Viale_GuessingModel} introduced the notion of Generic Laver Diamond at $κ$---which we denote $\Diamond_{\text{Lav}}(κ)$---asserting the existence of a single function from $κ\to H_κ$ that behaves much like a supercompact Laver function, except with generic elementary embeddings rather than internal embeddings. Viale proved that the Proper Forcing Axiom (PFA) implies $\Diamond_{\text{Lav}}(ω_2)$. We strengthen his theorem by weakening the hypothesis to a statement strictly weaker than PFA. We also show that the principle $\Diamond_{\text{Lav}}(κ)$ provides a uniform, simple construction of 2-cardinal diamonds, and prove that $\Diamond_{\text{Lav}}(κ)$ is quite prevalent in models of set theory; in particular: 1) $L$ satisfies $\Diamond_{\text{Lav}}^+(κ)$ whenever $κ$ is a successor cardinal, or when the appropriate version of Chang's Conjecture fails. 2) For any successor cardinal $κ$, there is a $κ$-directed closed class forcing---namely, the forcing from Friedman-Holy \cite{MR2860182}---that forces $\Diamond_{\text{Lav}}(κ)$.

math.LO↗