# THEMATIC PROGRAMS

May  2, 2016

THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES
20th ANNIVERSARY YEAR

July-December 2012 Thematic Program on Forcing and its Applications

## November 12-16, 2012 Workshop on Iterated Forcing and Large Cardinals

Organizing Committee:
Michal Hrusak, Paul Larson, Saharon Shelah, W. Hugh Woodin

### Abstracts

Joerg Brendle (KOBE University)
Methods in iterated forcing

We present some techniques for iterating forcing constructions.For example, we discuss Shelah's method of iterating by repeatedly taking ultrapowers of a forcing notion. We will also give a brief outline of Shelah's technique of iterating along templates. While we shall mention some applications, the focus will be on illustrating the basic ideas underlying these techniques.

Moti Gitik (Tel-Aviv University)
A weak generalization of SPFA to higher cardinals.

We apply a form of the Neeman iteration to finite structures with pistes. This allows to formulate a certain weak analog of SPFA for higher cardinals.

Martin Goldstern (Technische Universität Wien)
Cichon's diagram and large continuum

I will sketch a forcing construction of a model in which several well-known cardinal characteristics if the continuum (in particular: continuum itself, cofinality of null, uniformity of null, uniformity of meager, covering of meager) all have different values.
Joint work with Arthur Fischer, Kellner, Shelah. (Work in progress.)

John Krueger
Forcing with Models as Side Conditions

I describe a comparison of elementary substructures which allows for a uniform method of forcing with models as side conditions on $\omega_2$.

Heike Mildenberger (Albert-Ludwigs-Universität Freiburg)
Forcings with block sequences

I will discuss some new preservation theorems for forcings with block sequences.

A study of iterating semiproper forcing

I would like to introduce a way to iterate semiproper forcing. Suppose we have an initial segment, of limit length, of an iterated forcing. We consider the set of conditions that have sort of traceable countable stages. It turns out that this set of conditions forms a limit which sits between the direct and full limits. If we keep iterating semiproper p.o. sets under this limit, then every tail of the iteration is semiproper in the intermediate stage. In particular, the iteration itself is semiproper. This is a generalization of an iteration lemma on proper forcing under countable support.

Itay Neeman (University of California, Los Angeles)
Higher analogs of the proper forcing axiom

I will present a higher analogue of the proper forcing axiom, and discuss some of its applications. The higher analogue is an axiom that allows meeting collections of $\aleph_2$ maximal antichains, in specific classes of posets that preserve both $\aleph_1$ and $\aleph_2$.

This talk will include more details and proofs than my talk in the workshop on Forcing Axioms and their Applications. I will quickly survey the previous talk for audience members who were not present in the
previous workshop.

Ralf Schindler (WWU Münster)
An axiom.

We propose and discuss a new strong axiom for set theory.

Xianghui Shi (Beijing Normal University)
Some consequences of I0 in Higher Degree Theory

We present some consequences of Axiom I0 in higher degree theory. These results indicate some connection between large cardinals and general degree structures. We shall also discuss more evidences along this direction, raise some open questions. This is a joint work with W. Hugh Woodin.

Matteo Viale (University of Torino)
Absoluteness of theory of $MM^{++}$

Assume $\delta$ is a limit ordinal.
The category forcing $\mathbb{U}^\mathsf{SSP}_\delta$ has as objects the stationary set preserving partial orders in $V_\delta$ and as arrows the complete embeddings of its elements with a stationary set preserving quotient.

We show that if $\delta$ is a super compact limit of super compact cardinals and $\mathsf{MM}^{++}$ holds, then
$\mathbb{U}^\mathsf{SSP}_\delta$ completely embeds into a pre saturated tower of height $\delta$.

We use this result to conclude that the theory of $\mathsf{MM}^{++}$ is invariant with respect to stationary set preserving posets that preserve this axiom.