Abstracts for seminar talks at the KGRC

Meeri Kesälä, Finitary abstract elementary classes: Abstract

I will introduce the work in my Ph.D. thesis about Finitary abstract elementary classes and discuss some recent developments on the field.

More specifically, I will introduce and compare some frameworks in non-elementary model theory and discuss examples and topics such as categoricity transfer and independence calculus in these frameworks.

Philip Welch, In and around the Ramsey property: Abstract

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.

Vladimir Kanovei, Lebesgue measure and the coin-tossing game: Abstract

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.

Matteo Viale, Reflection principles and pcf theory: Abstract

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.

Radek Honzik, Easton theorem and large cardinals: Abstract

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)

Heike Mildenberger, A new forcing poset and an old question: Abstract

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.

Luca Motto Ros, Generalizations of Wadge degrees: Abstract

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.

Zhang Yi, A class of MAD families: Abstract

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.

Tapani Hyttinen, Geometric dependences in model theory: Abstract

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.

Martin Zeman, Combinatorial construction in extender models: What has been done: Abstract

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.

Agatha Walczak-Typke, A gentle introduction to non-structure of submodels of a large unstable homogeneous model, Part II: Abstract

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.

Katie Thompson, How to achieve Global Domination (in an inner model): Abstract

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.

Andrew Brooke-Taylor, Large cardinals and definable well-orders, Mk II: Abstract

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.

Thomas Johnstone, Indestructible cardinals and forcing axioms: Abstract

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.

Heike Mildenberger, There may be infinitely many near-coherence classes under u<d: Abstract

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.

Matteo Viale, A family of covering properties for forcing axioms and strongly compact cardinals, part 2: Abstract

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.

Matteo Viale, A family of covering properties for forcing axioms and strongly compact cardinals: Abstract

Andrew Brooke-Taylor, Quagmire forcing: Abstract

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.

Jakob Kellner, The Banach Mazur and pressing down games: Abstract

(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.

Andrew Brooke-Taylor, Large cardinals and definable well-orders: Abstract

I will show how, using Kurepa trees as oracles, one may perform a class forcing so that a generically chosen class of cardinals will be definable in the extension. In the extension model, we will then have a definable well order, GCH will hold, and any n-superstrong cardinals from the ground model will remain n-superstrong.

For further, light-hearted discussion, see http://www.logic.univie.ac.at/~andrewbt/DinosaurDWO.html

Richard Kaye, Nonstandard symmetric groups: Abstract

A nonstandard symmetric group is an internal finite symmetric group S_n inside a nonstandard model M of PA. This group can be considered internally as well as externally. Internally, it has a normal subgroup A_n of index 2, and A_n is simple for n>=5.

However, externally, A_n is an infinite group with an interesting normal subgroup structure. The main part of this talk will explain these normal subgroups of A_n. We conclude by examining interesting topological structures that can be imposed on these groups.

This is joint work with John Allsup, Birmingham.

Meeri Kesälä, Finitary Abstract Elementary Classes: Abstract

We know that first order logic fails to capture many natural classes of structures that appear in mathematics. Several generalizations of model-theoretical tools to non-elementary classes (Shelah 1985) is very general. We do not study structures in any specific language, but give axioms for an abstract elementary substructure-relation. However, if we want to study for example stability theory in this context, we may have to add some more specific assumptions for the class. We introduce finitary abstract elementary classes, a subclass of abstract elementary classes with many good properties. We also compare finitary classes to some other non-elementary classes by studying the behaviour of Galois types and category transfer. This is joint work with Tapani Hyttinen.

Martin Goldstern, All creatures great and small: Abstract

For any regular uncountable cardinal lambda I will describe a "creature-based" lambda^+-complete forcing notion that introduces a "wild" ultrafilter on lambda.
Assuming 2^lambda=lambda^+, we can find a sufficiently generic filter on this forcing notion; this allows us to construct a clone on lambda which is not contained in any coatom of the clone lattice, solving an old problem in clone theory. (These notions will be explained in the talk.)

I have sketched a corresponding result for lambda=omega in my talk in November 2002. Both results are a joint work with Saharon Shelah.

I will give a related talk (that concentrates on the algebraic rather than set-theoretic background) in the algebra seminar at TU Wien on April 7, 2006. http://www.dmg.tuwien.ac.at/fg1/seminar/20060407.html

Sebastiaan Terwijn, Randomness and relativization: Abstract

This will be an informal low-brow talk on some recent developments in recursion theory on random and generic sets. We will discuss several relations between randomness of finite strings and the theory of finite strings (Kolmogorov complexity). As it turns out, the notion of relativized computation plays a crucial role here. This is joint work with Andre Nies (Auckland) and Frank Stephan (Sydney).

Katie Thompson, Methods for solving universality problems: Abstract

We will discuss a number of ways of showing that universal models do or do not exist. The methods stem from model theory, set theory and category theory. We will see examples of these methods mostly using relational structures, but they can be applied to algebraic and topological structures as well. By comparing which methods work for different structures, one can find patterns in the behaviour of these structures with regard to universality.

Andres Caicedo, BPFA and the reals: Abstract

This is joint work with Boban Velickovic. I present some recent results (building on techniques introduced by Justin Moore) that allow us to code reals by ordinals in the presence of BPFA. I will also present some applications.

Jakob Kellner, A construction for non wellfounded forcing iterations, Part II: Abstract

I will show how to "countable-support-iterate" finitely splitting lim-sup tree forcings along arbitrary total orders. (Part of a joint work with S. Shelah called Saccharinity)

John Krueger, Combinatorial Principles Related to Adding Clubs: Abstract

A number of forcing posets have been defined which introduce a club subset to a given fat stationary subset of $\omega_2$ under various assumptions. I introduce a combinatorial property of $\omega_2$ which implies there exists a fat stationary subset of $\omega_2$ which cannot acquire a club subset by any forcing poset which preserves $\omega_1$ and $\omega_2$, answering a problem of Abraham and Shelah. This property follows from Martin's Maximum and is equiconsistent with a Mahlo cardinal.

Jakob Kellner, A construction for non wellfounded forcing iterations: Abstract

I will show how to "countable-support-iterate" finitely splitting lim-sup tree forcings along arbitrary total orders. (Part of a joint work with S. Shelah called Saccharinity)

Andres Caicedo, Preserving sequences of stationary subsets of omega_1: Abstract

Let M be an inner model that computes omega_1 correctly. We study whether we can find in M a partition of omega_1 into infinitely many sets that are stationary from the point of view of V.

Andrew Brooke-Taylor, Critical Points of Rank-to-Rank Embeddings: Abstract

One of the strongest large cardinal axioms we have posits the existence of an elementary embedding j from V_\lambda to V_\lambda for some limit ordinal \lambda. A peculiarity of it is that one such j will generate infinitely many more, not only through composition but also through the process of applying oneembedding to the graph of another.
I will talk about the structure generated in this way, and in particular the critical points of these embeddings.

Martin Goldstern, Continuous Fraisse conjecture and the number of Gödel logics: Abstract

Linear orders are naturally quasiordered by embeddability. Answering a question of Fraisse, Laver showed that this quasiorder, restricted to the scattered linear orders (those that do not contain a copy of the rationals), is a well-quasi-order; he also showed that there are exactly aleph1 equivalence classes (modulo bi-embeddability) of countable linear orders. In a joint paper with Arnold Beckmann and Norbert Preining we generalize this theorem to the natural quasiorder that is given by CONTINUOUS embeddability. A Gödel logic is given by a closed subset G of the unit intervall (containing 0 and 1). Fuzzy (relational) G-models are sets M with maps M^k -> G for every k-ary predicate symbol. A fuzzy satisfaction function is defined naturally; the "Gödel logic" associated with G is the set of all sentences which have value 1 in every fuzzy G-model. All these logics are contained in the set of classical validities; as an application of the continuous Fraisse conjecture, we show that there are only countably many Gödel logics.

Andres Caicedo, Bounded forcing axioms and projective well-orderings of the reals: Abstract

In the absence of Woodin cardinals, fine srtuctural inner models for mild large cardinal hypotheses admit set forcing extensions where bounded forcing axioms hold and the reals are projectively well-ordered.

Grzegorz Plebanek, Measures defined on sigma algebras contained in Bor[0,1]: Abstract

The talk is devoted to properties of measures defined on sigma algebras contained in Bor[0,1], or more generally, in Bor(X), where X is a Polish space. In particular, we are going to mention some open problems on measures and infinite games.

John Krueger, Strong Compactness and Stationary Sets 2: Abstract

I will show how to construct a model in which $\kappa$ is a strongly compact cardinal and the set $S(\kappa,\kappa^+) = \{ a \in P_\kappa \kappa^+ : \ot(a) = (a \cap \kappa)^+ \}$ is non-stationary.

John Krueger, Strong Compactness and Stationary Sets: Abstract

I will show how to construct a model in which $\kappa$ is a strongly compact cardinal and the set $S(\kappa,\kappa^+) = \{ a \in P_\kappa \kappa^+ : \ot(a) = (a \cap \kappa)^+ \}$ is non-stationary.

James Hirschorn, CCC Forcing and Splitting Reals, Part 2: Abstract

Prikry asked if it is relatively consistent with the usual axioms of ZFC that every nontrivial ccc forcing adds either a Cohen or a random real. Both Cohen and random reals have the property that they neither contain nor are disjoint from an infinite set of integers in the ground model, i.e. they are splitting reals. In this note I show that it is relatively consistent with ZFC that every non atomic weakly distributive ccc forcing adds a splitting real. This holds, for instance, under the Proper Forcing Axiom and is provided using the P-ideal dichotomy first formulated by Abraham and Todorcevic and later extended by Todorcevic. In the process, I show that under the P-ideal dichotomy every weakly distributive ccc complete Boolean algebra carries a Maharam submeasure, a result which has some interest in its own right. Using a previous theorem of Shelah it follows that a modified Prikry conjecture holds in the context of Souslin forcing notions, i.e. every non atomic ccc Souslin forcing either adds a Cohen real or its regular open algebra is a Maharam algebra.

James Hirschorn, CCC Forcing and Splitting Reals: Abstract

Andres Caicedo, CH and the saturation of the nonstationary ideal on omega_1: Abstract

The saturation of the nonstationary ideal on omega_1 was shown consistent (with ZFC) from a strong form of determinacy by Steel and VanWesep in the early 80's. Their techniques produced a model where CH fails. It has been an open question since then whether a model can be produced where the ideal is saturated and CH holds. Although this problem is still open, significant progress towards a (negative) solution was made by Woodin in the 90's. Specifically, Woodin proved that the saturation of the ideal contradicts CH, *in the presence of large cardinals*. In fact, a definable counterexample is produced. However, no such definable counterexample can exist if the large cardinals are absent from the picture, and apparently a completely new idea is necessary to settle the problem in this case. A nice side effect of Woodin's techniques is the development of the theory of P_max.

In this talk I plan to present Woodin's result, together with its limitations.

Menachem Kojman, (KGS Lecture) Topology and Combinatorics of Singular Cardinals: Abstract

In the talk I will present the pcf approach to singular cardinals and some of its applications to topology and to combinatorics.

Akihiro Kanamori, Zermelo and Set Theory: Abstract

Ernst Friedrich Ferdinand Zermelo (1871--1953) transformed the set theory of Cantor and Dedekind into {\it abstract} set theory in the first decade of the 20th century by incorporating the Axiom of Choice and providing a simple and workable axiomatization, and thereby tempered the ontological thrust of early set theory and established the basic conceptual framework for the development of modern set theory. Two decades later Zermelo promoted a distinctive cumulative hierarchy view of models of set theory that would have a modern resonance. In this paper Zermelo's published mathematical work in set theory is described and analyzed in its historical context, with the hindsight afforded by the knowledge of what has endured in the subsequent development of set theory. Much can be and has been written about philosophical and biographical issues and about the reception of the Axiom of Choice, and we will refer and defer to others, staying the course through the decidedly mathematical themes and details.

Jakob Kellner, Preserving Preservation 2: Abstract

We investigate under which conditions preservation of positivity is iterable (joint work with S. Shelah).

Jakob Kellner, Preserving Preservation: Abstract

We investigate sufficient conditions that allow to preserve (in limit setps of countable support interations) the property of preservation of positivity.

Andres Caicedo, Well-Orderings of the Reals and Real-valued Measurability: Abstract

I will prove that (if there are measurables, then) there are extensions of V where the continuum is real-valued measurable and there is a Sigma-2-2 well-ordering of the reals. I will also show that natural strengthenings of the concept of real-valued measurability imply that the reals are not Sigma-2-n well-orderable for any n (so the result is not vacuously true) and that real-valued measurability implies that no well-ordering of the reals belongs to L(R) (so the result is non-trivial.) Depending on time, I might mention what is known with respect to real-valued measurability of the continuum and Sigma-2-1 well-orderings, which is a much more complicated story.

Andrzej Roslanowski, Around Sheva-Sheva-Sheva: forcing for the lambda-reals (where lambda is inaccessible): Abstract

A number of cardinal characteristics related to the Baire space, the Cantor space and/or the combinatorial structure of [omega]^omega can be extended to the spaces obtained by replacing omega by lambda (for any infinite cardinal lambda). Following the tradition of Set Theory of the Reals we may call cardinal numbers defined this way "cardinal characteristics of lambda-reals". The menagerie of those characteristics seems to be much larger than the one for the continuum, but to decide if the various definitions lead to different (and interesting) cardinals we need a well developed forcing technology. We will present initial steps in this direction. This is a joint work with Saharon Shelah and the paper is available at http://arxiv.org/math.LO/0210205

Jakob Kellner, Preserving non-null with transitive nep forcings: Abstract

I will introduce the definitions of Suslin ccc, Suslin proper and transitive nep, demonstrate that many of the usual "definable" forcings of sets of reals are Suslin+, and present an application of these notions, a simplified version of Shelah's "preserving a little implies preserving much": If I is an ideal generated by a Suslin ccc forcing (e.g. null or meager), and P is a transitive nep forcing, and (in V and every forcing extension) P forces that no old positive Borel-set becomes null, then P forces that no old positive set becomes null.

Ralf Schindler, Sharps, pistols, and the \Sigma^1_3 correctness of K: Abstract

I'll give a simple proof of the following result of Steel and Welch: Let x be a solution to the \Pi^1_2 property \Phi(-), and suppose that x^#, x^##, x^###, etc. exist; suppose further that 0^pistol doesn't exist. Then K contains a solution to \Phi(-).

Ralf Schindler, A combinatorial proof of \Sigma^1_3 correctness of K: Abstract

I'll show you a new, simple, and purely combinatorial proof of the following result, which is originally due to Steel (1993): Let the real x be a solution to the \Pi^1_2 property \Phi(-), and suppose that x^dagger exists; suppose further that there's no inner model with a Woodin cardinal. There is then a lightface mouse which contains a solution to \Phi(-).

Arnold Beckmann, Some words on well-foundedness principles over definable sets: Abstract

Well-foundedness principles are used in the definition of proof theoretic ordinals. If the 2nd order quantification in this definition is restricted to definable sets (e.g. arithmetical sets in case of Peano Arithmetic) we obtain "pathological" proof theoretic ordinals by a result of Kreisel. Does the situation differ for other pairs of theories and definable sets? Or, is there a way out of this misery?

Lorenz Halbeisen, Partition-ultrafilters: Abstract

The talk will be about some topological properties of the space of partition-ultrafilters as well as about some special partition-ultrafilters, like a partitioned version of Ramsey ultrafilters.

Mladen Pavicic, Classical logic models: Abstract

A well-known ortholattice model of classical propositional logic is the Boolean algebra (a distributive ortholattice, which is therefore orthomodular as well). In this talk I will show that there is also another ortholattice model of classical propositional logic which is neither distributive nor orthomodular so that classical propositional logic turns out to be non-categorical. I give the soundness and completeness proofs for the new model and compare them with those for the Boolean algebra.

Tomek Bartoszynski, Perfectly meager sets: Abstract

A set of reals X is perfectly meager if X is meager inside every perfect set P. Uncountable perfectly meager sets can be constructed in ZFC. In 1935 Marczewski asked if the product of perfectly meager sets is perfectly meager. In this talk I will discuss the answer to this question given by the following two theorems.

Theorem (Reclaw 1990) "No" is consistent with ZCF.

Theorem (T.B. 2000) "Yes" is consistent with ZCF.

Martin Goldstern, Clones on regular cardinals: Abstract

A clone on a set X is a set of functions (on any finite arity) which is closed under composition (e.g., f(x,y) and g(x,y) are in the clone, then also g(f(x,z), f(z, y)) is in the clone.) The set of clones on X forms a complete algebraic lattice. I will present some results about this lattice for two cases:

|X| is a weakly compact cardinal (or aleph_0)
|X| is a successor of a regular cardinal. In this case we can use a strong negative partition relation to get a "nonstructure" result.

(These are results from a joint paper with Shelah, GoSh:747)

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