C. Switzer

Baumgartner's Axiom and Small Posets

Pedro Marun, Saharon Shelah, Corey Bacal Switzer

We contribute to the study of $ℵ_{1}$ -dense sets of reals, a mainstay in set theoretic research since Baumgartner's seminal work in the 70s. In particular, we show that it is co…

Adding $ℵ_{ω}$ many Cohen reals

Pedro Marun, Saharon Shelah, Corey Bacal Switzer

Abstractly, the generic extensions after $ℵ_{ω}$ -many Cohen reals and $ℵ_{ω + 1}$ -many Cohen reals must be different for reasons of uniform density the relevant Boolean algeb…

Eventual Capture on a Measurable Cardinal

Tom Benhamou, Corey Bacal Switzer

We continue the study from \cite{BrendleFreidmanMontoya, vandervlugtlocalizationcardinals} of localization cardinals $\mfb_κ(\in^*)$ and $\mfd_κ(\in^*)$ and their variants at regul…

A note on adding isomorphisms and the pseudointersection number

Corey Bacal Switzer

We prove that for every tower $T$ there are $ℵ_{1}$ -dense $A$ and $B$ so that any ``reasonable" forcing notion $P$ -- an adjective that includes all known ones…

Variants of Baumgartner's Axiom for Lipschitz Functions on Baire and Cantor Space

Corey Bacal Switzer

We consider several variants of Baumgartner's axiom for $ℵ_{1}$ -dense sets defined on the Baire and Cantor spaces in terms of Lipschitz functions with respect to the usual metri…

Weak Baumgartner axioms and universal spaces

Corey Bacal Switzer

If $X$ is a topological space and $κ$ is a cardinal then $BA_{κ} (X)$ is the statement that for each pair $A, B \subseteq X$ of $κ$ -dense subsets there is an autohomeomorphis…

Reflection Properties of Ordinals in Generic Extensions

Juan P. Aguilera, Corey Bacal Switzer

We study the question of when a given countable ordinal $α$ is $Σ_{n}^{1}$ - or $Π_{n}^{1}$ -reflecting in models which are neither $PD$ models nor the constructible universe, focus…

Separating Subversion Forcing Axioms

Hiroshi Sakai, Corey Bacal Switzer

We study a family of variants of Jensen's\emph{subcomplete forcing axiom}, $SCFA$ and \emph{subproper forcing axiom}, $SubPFA$ . Using these we develop a general t…

Generic Selective Independent Families

Vera Fischer, Corey Bacal Switzer

We prove that the generic maximal independent family obtained by iteratively forcing with the Mathias forcing relative to diagonalization filters is densely maximal. Moreover, by c…

Filters, ideal independence and ideal Mrówka spaces

Serhii Bardyla, Jonathan Cancino-Manríquez, Vera Fischer +1

A family $A \subseteq [ω]^{ω}$ such that for all finite ${X_{i}}_{i \in n} \subseteq A$ and $A \in A ∖ {X_{i}}_{i \in n}$ , the set $A \setminus \big…

Cohen Preservation and Independence

Vera Fischer, Corey Bacal Switzer

We provide a general preservation theorem for preserving selective independent families along countable support iterations. The theorem gives a general framework for a number of re…

Filters and Ideal Independence

Jonathan Cancino-Manríquez, Vera Fischer, Corey Bacal Switzer

A family $I \subseteq [ω]^{ω}$ such that for all finite ${X_{i}}_{i \in n} \subseteq I$ and $A \in I ∖ {X_{i}}_{i \in n}$ , the set $A \setminus \big…

The Special Tree Number

Corey Bacal Switzer

Define the special tree number, denoted $st$ , to be the least size of a tree of height $ω_{1}$ which is neither special nor has a cofinal branch. This cardinal had previou…