2007-01-11Jakob Kellner (KGRC, The Hebrew University of Jerusalem)The Banach Mazur and pressing down games
(Joint work with Matti Pauna and Saharon Shelah)

I will compare the pressing down game and the Banach Mazur game and show that they can be different on S21.

2007-01-18Sy David Friedman (KGRC)Consistency completeness
2007-01-25Andrew Brooke-Taylor (KGRC)Quagmire forcing
When trying to preserve large cardinals while doing class forcing, a standard trick is to obtain a "mastercondition" - a single condition that the generic must lie below to guarantee that the large cardinal is preserved. If the forcing is homogeneous enough, this choice of an appropriate generic can be performed "after the fact", in the extension V[G] by any generic.

However, the standard forcing to give morasses does not enjoy this sort of homogeneity. We shall show how to modify it so that it does, and in doing so, produce morasses with an extra property.

2007-03-15Sy-David Friedman (KGRC)An easier proof of Woodin's Theorem
2007-03-22Matteo Viale (KGRC)A family of covering properties for forcing axioms and strongly compact cardinals
I introduce a simple device to investigate the combinatorics of singular cardinals above a strongly compact or assuming strong forcing axioms. In particular I obtain an elementary proof of SCH from PFA and several constraints on the possible scenarios to change cofinalities while preserving forcing axioms or strongly compact cardinals.

2007-03-29Matteo Viale (KGRC)A family of covering properties for forcing axioms and strongly compact cardinals, part 2
See [[Talks/2007/03-29]]
2007-04-19Heike Mildenberger (KGRC)There may be infinitely many near-coherence classes under u<d
We show that in the models of u<d from Blass and Shelah there are infinitely many near-coherence classes of ultrafilters, thus answering a question by Banakh and Blass in the negative.

2007-04-26Sy-David Friedman (KGRC)Infinite Time Turing Machines
2007-05-03Agatha Walczak-Typke (KGRC)A gentle introduction to non-structure of submodels of a large unstable homogeneous model, Part I
See [[Talks/2007/05-03]]
2007-05-10Thomas Johnstone (City University of New York)Indestructible cardinals and forcing axioms
Determining which cardinals can be made indestructible by which classes of forcing has been a major interest in modern set theory. Inspired by Laver's celebrated result for supercompact cardinals, I will present a method of making strongly unfoldable cardinals indestructible. These cardinals strengthen weakly compact and indescribable cardinals, yet they are rather small in the hierarchy of large cardinals, as they are consistent with V=L. Starting with a strongly unfoldable cardinal kappa, I will produce a forcing extension V[G], in which the strong unfoldability of kappa is indestructible by all <kappa-closed, kappa+ preserving posets. In particular, the weak compactness and indescribability of kappa is indestructible. Previously known results would have had to assume the existence of a strong or supercompact cardinal to obtain this general indestructibility. Combining the method with the idea of Baumgartner's proof of the relative consistency of the Proper Forcing Axiom PFA, I will establish the consistency of a weakening of PFA relative to the existence of a strongly unfoldable cardinal. I will also discuss several related open questions. Part of the material in this talk is joint work with Joel David Hamkins.

2007-05-24Andrew Brooke-Taylor (KGRC)Large cardinals and definable well-orders, Mk II
This will be an entirely revamped version of the talk I gave last December. Instead of Kurepa trees, we now code using the existence of diamond star sequences. We also broaden the range of large cardinals to be preserved, and give a more detailed discussion of how close we can come to preserving all cardinals of a given kind. Finally, if there's time left at the end, I'll talk briefly about something completely different that will also appear in my thesis: universal morasses.

2007-05-31Katie Thompson (KGRC)How to achieve Global Domination (in an inner model)
Cummings and Shelah developed a generalised notion of the dominating number and used a non-linear iteration of Hechler forcing to fix the dominating number for lambda and 2^lambda for all regular lambda with minimal restrictions. We would like to find an inner model for this global property, but the techniques available for finding inner models assuming only 0# cannot be used with this forcing.

Therefore, in joint work with Sy-David Friedman, we restrict ourselves first to finding an inner model of Global Domination, a global property where the dominating number is less than 2^lambda for all regular lambda. Using perfect tree forcing Friedman and I get Global Domination in an inner model for inaccessible cardinals. We would like to extend this to all regular cardinals by sneaking in some Hechler forcing at successors, but run into problems with the mix of forcings at the successors of inaccessibles. The solution has a lot in common with making chocolate mousse.

2007-06-14Agatha Walczak-Typke (KGRC)A gentle introduction to non-structure of submodels of a large unstable homogeneous model, Part II
The work presented is joint with S-D Friedman and T Hyttinen. We aim to generalize a very nice result of Friedman, Hyttinen, and Rautila, which ties first-order model theoretic classification theory to constructibility under the assumption of 0#, to a non-elementary model theoretic setting. The orignal result stated:
''Theorem.'' Assume 0# exists and let T be a constructible first-orer theory which is countable in the constructible universe L. Let \kappa be a cardinal in L larger than (\aleph_1)^L. Then the collection of constructible pairs of models A,B of T, |A|,|B|=\kappa, which are isomorphic in a cardinal- and real-preserving extension of L is itself constructible if and only if T is classifiable (i.e. superstable with NDOP and NOTOP).
We have chosen Homogeneous Model Theory as a good setting for generalizing this result.

In Part I of this talk, a gentle introduction to Homogeneous Model Theory was given, as well as a justification as to why this is a good setting to choose.

In Part II, one easy step for our generalization will be sketched: the unstable case.

2007-06-21Martin Zeman (UC Irvine)Combinatorial construction in extender models: What has been done
Combinatorial constructions in higher extender models are important for at least two reasons. First, they give us detailed information both on combinatorial principles and canonical models for large cardinals. Second, they give rise to new inner model theoretic techniques and enable us to see inner models from new aspects that are interesting in their own right. I will summarize known combinatorial constructions, show differences between them and try to explain what would be the next direction of research in this area.

2007-06-28Tapani Hyttinen (University of Helsinki)Geometric dependences in model theory
I call a dependence relation geometric if the dependences between any two sequences are determined by dependences between finite subsequences. In the talk I will give a short introduction to the theory of geometric dependences. A special attention is given to dependences in the context of abstract elementary classes and to examples.

2007-10-04Zhang Yi (Sun Yat-sen University, China)A class of MAD families
I will introduce a class of mad families which naturally arised from several branches of mathematics. I will talk about possible order relationships between these families and others. Moreover, I will introduce several open problems which I have been working on for quite a long time.

2007-10-11Luca Motto Ros (KGRC)Generalizations of Wadge degrees
I will give a brief history of some of the most important results in the Wadge's theory and survey some recent developments about general reducibilities for sets of reals.

2007-10-18Moti Gitik (Tel Aviv University)Some remarks about precipitous ideals
2007-10-25Heike Mildenberger (KGRC)A new forcing poset and an old question
We introduce a highly undefinable notion of forcing and we analyse it by developing variations of the theorems of Hindman and Milliken and Taylor. We use the forcing to answer an old question.

2007-11-08Radek Honzik (KGRC)Easton theorem and large cardinals
The continuum function F on regular cardinals is known to have great freedom - that is providing we do not mind destroying some large cardinals. If we wish to preserve for instance measurable cardinals and realize F, some restrictions must be put on F (for instance GCH cannot first fail at the given measurable cardinal). We show that if we put some very mild restrictions on F, measurable cardinals will be preserved in some generic extension realizing F.

(This work is joint with Sy D. Friedman)

2007-11-15Matteo Viale (KGRC)Reflection principles and pcf theory
We present some application of reflection principles to the analysis of the partial order of reduced product of regular cardinal. The guiding example being the study of the partial order (\prod_n\aleph_n,<^*), where f<^*g if for finitely many n f(n)\geq g(n). The main original result is that a reflection principle on \aleph_2 which is equiconsistent with \aleph_2 being weakly compact in L and which follows from Martin's maximum implies that club many points of cofinality \aleph_2 below \aleph_{\omega+1} are approachable. This is obtained by a combination of two theorems: one by me and the other by Assaf Sharon. We also apply this result to deny many instances of Chang conjectures.
The first seminar will be an introduction to the subject. In the second one we will focus on the new results.

2007-11-22Matteo Viale (KGRC)Reflection principles and pcf theory, part II
See [[Talks/2007/11-22]]
2007-11-29Vladimir Kanovei (Institute for Information Transmission Problems, Moscow)Lebesgue measure and the coin-tossing game
Given a set A of infinite dyadic sequences, we consider a game between G, the gambler, and C, the casino. C successively plays bits b_0,b_1,b_2,... , and C definitely loses if the infinite sequence b=<b_0,b_1,b_2,...> does NOT belong to A.
And G bets on every next move of C.
Beginning with the initial balance say \(1, G can bet any amount less than the current balance on one of two possible moves of C (0 or 1), and if C makes that move then the balance accordingly increases by the amount of bet.
Otherwise the balance decreases.
The final outcome of the game can be defined in terms of the limit of the supremum of the balance values.
And it turns out that the existence of certain strategies for G and C characterizes the Lebesgue measure characteristics of the set A. In brief, the smaller A is the bigger gains Casino can guarantee.

2007-12-06Radek Honzik (KGRC)Easton's theorem and large cardinals, Part 2
See [[Talks/2007/12-06]]
2007-12-13Philip Welch (University of Bristol)In and around the Ramsey property
Recent work on the Mutual Stationarity property has prompted looking at some finite sequence "mutual stationarity" of subsets of omega_1 and omega_2. We discuss some joint work with I. Sharpe on this, related also to mild strengthenings of the Chang Property; some further topics in the Jonsson/Ramsey hierarchy may be mentioned if time permits.