# Prikry Forcing Online

On December 14th we will have a one day online workshop on the topic of Prikry forcing, with the focus on early career speakers. The meeting will take place online, as appropriate for the year 2020, and more details will follow soon.

The main focus of the meeting will be Prikry forcing, but we will have rump sessions with 10 minutes talks open for any young set theory researcher interested in presenting any result in set theory. We encourage everyone to apply as long as there's room for it (the list of speakers will be updated as we go along)!

## Schedule (GMT)

 10:00–10:50 Alejandro Poveda (video) 11:00–11:50 Tom Benhamou (video) 12:00–12:40 Rump session #1 Kenta Tsukuura (video) Zhixing You (video) 12:45–14:00 Break 14:00–14:50 Sittinon Jirattikansakul (video) 15:00–15:50 Chris Lambie-Hanson (video) 16:00–17:00 Rump session #2 Richard Matthews (video) Monroe Eskew (video) Elliot Glazer (video)
In the rump sessions each talk is 15 minutes and will have an additional 5 minutes for questions at the end.
Main Talks
Alejandro Poveda
A new iteration scheme with applications to singular cardinals combinatorics (slides)
In this talk we will give an overview of the theory of $\Sigma$-Prikry forcings and their iterations. We will begin motivating the class of $\Sigma$-Prikry forcings and showing that many Prikry-type posets that center on countable cofinalities fall into this framework. Afterwards, we will present a viable iteration scheme for this family. Finally, and if time permits, we will try to discuss some applications of the framework to the investigation of consistency result in singular combinatorics. Specifically, we shall discuss a recent construction of a model where GCH holds below $\aleph_\omega$, the Singular Cardinal Hypothesis fails at $\aleph_\omega$ and every stationary subset of $\aleph_{\omega+1}$ reflects. This shows the mutual consistency of two classical results by Magidor from 1977 and 1982, respectively. This is a joint work with Assaf Rinot and Dima Sinapova.
Tom Benhamou
Intermediate models of Prikry-type forcings (slides)
It is well known that any intermediate model of a Cohen forcing or random real forcing extension is a Cohen forcing or random real extension respectively. Recently, this phenomena was proved by Gitik, Koepke, and Kanovei to hold also for the standard Prikry forcing with a normal ultrafilter, i.e. every intermediate model of a Prikry generic extension is again Prikry generic for the same ultrafilter. In this talk we will ad-dress more complicated Prikry type forcings such as Magidor–Radin forcing, the tree-Prikry forcing, and present several results regarding intermediate models of generic extensions of these forcings.
Sittinon Jirattikansakul
Blowing up the power of cardinals which are singular in the ground models with collapses (slides)
In this talk, we introduce a new kind of extender-based forcings. Given a singular cardinal which is a limit of large cardinals, new Gitik's forcings allow us to blow up the power of that singular cardinal. We will present the features of Gitik's forcings with interleave collapses, and sketch the ideas on how the proof of the Prikry property works. Time permitting, we will discuss how we derive a scale, which is a particular object used to violate the SCH.
Chris Lambie-Hanson
Pseudo-Prikry sequences (slides)
Prikry-type forcing notions have been a uniquely effective tool for proving consistency results about singular cardinal combinatorics. In this talk, we will survey a number of results giving some indication as to why this is the case. In particular, we will show that, in a wide variety of circumstances in which $V\subseteq W$ are models of set theory such that there is at least one regular cardinal of $V$ that is singular in $W$ (or such that certain PCF-theoretic relations hold between the models), there provably exists an object in $W$ that approximates a generic object over $V$ for some Prikry-type forcing notion. We will begin the talk by discussing the earliest work in this direction, done by Gitik and by Džamonja and Shelah in the 1990s, and then turn to more recent work by Magidor and Sinapova, Gitik, and the speaker.
Mini Talks
Kenta Tsukuura (Tsukuba) Prikry forcing and universal collapse (slides)
It is still open whether strongly saturated ideal can exist on the successors of singular cardinals or not. To solve this, we study a universal collapse combined with a Prikry forcing and give a saturated ideal which is explicit.
Zhixing You (Chinese Academy, Barcelona) Minimal Magidor-type forcing (Countable case) (slides)
For any countable limit ordinal $\delta$, we constructed a Minimal Magidor-type forcing, which adds an increasing continuous sequence $C_G$ of length $\delta$, such that any intermediate extension is of the form $V[C_G\restriction\alpha]$, where $\alpha\leq\delta$ is limit.
Richard Matthews (Leeds) Big Classes and Class Forcings (slides)
A proper class is "big" if it surjects onto any non-zero ordinal. We shall study this notion in subsystems of ZFC, particularly without power set. As an approach, we shall look at Collection principles in symmetric submodels of class forcings.
Monroe Eskew (Vienna) Changing tail types (slides)
We very briefly describe a diagonal Prikry forcing designed to get Chang’s Conjecture between as many pairs of singulars as possible. It allows us to take a successor of a singular of type $\aleph_{\alpha+\omega}$ and change it into the successor of one of longer tail type. The open problem is on how long an interval of cardinals can this be done.
Elliot Glazer (Harvard) Paradoxes of perfectly small sets (slides)
We refute from a modest choice principle Freiling's axiom of symmetry for perfectly small subsets of $[0,1]$. Here "perfectly small" means "has perfectly many disjoint translates.

## Registration

If you wish to attend please let us know by filling the following form. If you want to contribute a talk, please register by December 10th.