2008-01-10Jörg Brendle (Kobe)Malyhin's Problem
2008-01-31Meeri Kesälä (KGRC)Finitary abstract elementary classes
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.

2008-03-06Sy-David Friedman (KGRC)Some natural equivalence relations in the Solovay model
2008-03-13Bernhard Irrgang (Universität Bonn)A remark on a theorem by Hajnal and Juhasz
By a theorem of Hajnal and Juhasz, \(card(X)\leq exp(exp(spread(X)))\) holds for all Hausdorff spaces \(X\). Juhasz asked, if the second \(exp\) is necessary. I will present a result which shows that \(card(X)=exp(exp(spread(X)))\) for a \(T_3\) space \(X\) with spread \(\omega_1\) is consistent. Previously, Fedorcuk constructed from \(\diamondsuit\) a \(T_3\) space \(X\) with spread \(\omega\) and \(card(X)=exp(exp(spread(X)))\).

2008-04-03Sebastiaan Terwijn (KGRC)Reductions by computable functionals
Turing reducibility is a central notion from computability theory measuring the relative computability of reals: A real A reduces to a real B if there is a computable functional mapping B to A. We can also use computable functionals to reduce sets of reals to each other. This gives rise to two structures generalizing the Turing degrees: The Medvedev and the Muchnik lattices. In this talk we will discuss some structural properties of these lattices.

2008-04-10Heike Mildenberger (KGRC)Combinatorics related to the P-ideal dichotomy
Lyubomyr Zdomskyy showed: Under the P-ideal dichotomy for ideals with aleph_1 generators together with some more relatively mild conditions, certain stronger forms of L-spaces do not exist. In order to see how this is related to work by Kunen and by Szentmikl\'{o}ssy, we investigate whether Martin's Axiom for aleph_1 dense sets necessarily holds in these models.

2008-04-17Luca Motto Ros (KGRC)Good Borel reducibilities for sets of reals
We will present an (almost) complete analysis (under AD) of Borel reducibilities using a quite general approach, and we will give various examples. In particular, we will show that the degree-structure induced by a good Borel reducibility is completely determined by its characteristic set (which correspond to the first nontrivial level of the corresponding hierarchy), and we will prove a dichotomy theorem for these structures. If time permits, we will also briefly describe the Martin-Monk method, which is one of the main tools involved in this theory.

2008-04-24Menachem Magidor (Hebrew University, Jerusalem)Skolem-Löwenheim Cardinals for Generalised Logics
2008-05-08Ernest Schimmerling (Carnegie-Mellon University)Some open questions about L
There are indeed open questions about L. I'll state a few and describe recent related results on two topics: forcing axioms and mutual stationarity.

2008-05-15Juliette Kennedy (University of Helsinki)Square Principles and Model Theory
We visit some facts and questions about models of arithmetic inside reduced products, and the more general set-theoretic/model-theoretic results they have led to. A finitary square principle equivalent (in certain important cases) to the universality of an associated reduced product is introduced. This is also equivalent to an isomorphism theorem for ultrapowers of elementarily equivalent models. The principle follows from a cardinal arithmetic assumption, and can be shown to fail under the assumption of strongly compact cardinals. This gives the independence of the model-theoretic principles, under a large cardinal assumption. This is joint work with Saharon Shelah and Jouko Vaananen.

2008-05-29Martin Koerwien (University of Illinois at Chicago)Omega-stability, Borel reducibility and Scott height
2008-06-05Sy-David Friedman (KGRC)The number of normal measures
2008-06-12Meeri Kesälä (KGRC)Finitary AEC and infinitary languages
I will prove the recent result by David Kueker, that finitary AEC are closed under L_infinity_omega equivalence. This implies several definability results for finitary abstract elementary classes and also raises some open questions.

The proof does not use advanced model theory, but countably closed and unbounded sets and the cub-game, which was introduced by Kueker in 1977. The proof resembles the interplay between cub sets and the axioms for abstract elementary classes with countable Löwenheim-Skolem number. We also explain the role of the property finite character.

2008-06-19James Cummings (Carnegie Mellon University)Strong reflection
(joint work with Dorshka Wylie) Let \(S, T\) be stationary subsets of an uncountable regular \(\kappa\). Say that \(S <^* T\) iff every stationary subset of \(S\) reflects at almost every point in \(T\). Let \(S^n_j = \{ \alpha < \omega_n : cf(\alpha) = \omega_j \}\). We construct a model with a sequence of stationary sets \(B_n \subseteq S^{n+2}_{n+1}\) for \(n < \omega\), such that \(S^{n+2}_i <^* B_n\) for all \(i \le n\), in which the sets \(B_n\) are as large as possible (it is not possible that \(B_n = S^{n+2}_{n+1}\)).

2008-06-26Alessandro Andretta (Torino)Definable cardianalities of boldface pointclasses
The question: "What is the size of a boldface pointclass?" is trivial under the Axiom of Choice, but becomes very interesting if we restrict ourselves to the realm of definable sets and maps. Assuming the Axiom of Determinacy a complete description of the cardinality of the boldface pointclasses is given, showing that the size of a pointclass is tightly related to its descriptive-set-theoretic comlexity.

This is joint work with G. Hjorth and I. Neeman.

2008-10-02Sy-David Friedman (KGRC)Internal Consistency
A statement is internally consistent iff it holds in an inner model, assuming the existence of inner models with large cardinals. This notion gives rise to a new type of relative consistency result:

"""ICon(ZFC + LC) --> ICon(ZFC + S)"""

where """ICon""" denotes "internally consistent", "LC" stands for some large cardinal hypothesis and """S""" is a statement of set theory. Internal consistency results demand techniques beyond those used for ordinary consistency results and come in two types. Type 1 results are those where "LC" is taken to be "0# exists". In this case, the methods of generic modification (F-Ondrejovic) and partial master conditions (F-Thompson) are typically used. Type 2 results arise when a statement can be shown by forcing to consistently hold in """V_kappa""" where """kappa""" is a measurable cardinal. In this second case, the methods used are typically generic modification (Woodin) or various tree-forcing methods (F-Thompson for Sacks products, Dobrinen-F for Sacks iterations and F-Zdomskyy for Miller iterations). In this talk I'll discuss internal consistency in the contexts of cardinal exponentiation, global domination, the tree property, embedding complexity and the cofinality of the symmetric group.

2008-10-09Vera Fischer (KGRC)The consistency b = kappa ^lt; s = kappa^+
In 1984 S. Shelah obtains the consistency of b = omega_1 < s = omega_2 using a proper forcing notion of size continuum, which adds a real not split by the ground model reals and satisfies the almost omega^omega-bounding property. We obtain a sigma-centered suborder of Shelah's poset, which behaves very similarly to the larger forcing notion: it adds a real not split by the ground model reals and preserves the unboundedness of a chosen unbounded, directed family of reals. Thus under an appropriate finite support iteration of length kappa^+, where kappa is an arbitrary regular uncountable cardinal, we obtain the consistency of b = kappa < s = kappa^+

2008-10-16Ekaterina Fokina (KGRC)Computable uncountably categorical structures
Computable model theory hybridizes two branches of mathematical logic, namely, model theory and computability (recursion) theory. Given a suitable way to code mathematical objects into natural numbers, we can talk about effective properties of these objects. The main questions of computable model theory are the following. Let T be a first-order theory. Does T have a computable (recursive) model? If T has a computable model, what is the algorithmic complexity of T? Can we say something about the complexity of other models of T? Such questions are especially interesting for objects that are important from the model-theoretic point of view. In this talk we will discuss the known results and open questions for the class of uncountably categorical theories.

2008-10-23Asger Törnquist (KGRC)Automorphisms of S_infty/fin
Let S_infty be the full permutation group on \omega and let fin denote the (normal) subgroup of permutations that act identically on a co-finite set. In this talk I will discuss some results about the structure of Aut(S_\infty/fin), analogous to the well known results by Rudin, Shelah and Velickovic about Aut(P(\omega)/fin). Specifically, I will show that all "definable" automorphisms are implemented by a bijection between co-finite sets, but that CH allows us to construct one that is not.

2008-11-06Lyubomyr Zdomskyy (KGRC)Measurable cardinals and the Cofinality of the Symmetric Group
Assuming the existence of a hypermeasurable cardinal, we shall construct a model of Set Theory with a measurable cardinal \(kappa\) such that \(2^kappa=kappa^++\) and the group \(Sym(kappa)\) of all permutations of \(kappa\) cannot be written as a union of a chain of proper subgroups of length \(<kappa^++\). The proof involves the iteration of a suitably defined uncountable version of the Miller forcing poset as well as the ``tuning fork'' argument introduced by S.D. Friedman and K. Thompson.

Based on the joint work with S.D. Friedman

2008-11-13Philip Welch (University of Bristol)Lower Bounds for consistency of  two substructure properties at aleph_omega
We consider algebras A on aleph_omega, and look at possible mutual stationarity properties they may enjoy. ("Mutual stationarity" is a concept introduced by Foreman and Magidor in their discussion of the, generally, non-saturatedness of the non-stationary ideal.) We look to see whether a sequence of sets S_n each stationary below aleph_n+1 can be "simultaneously stationary" as a sequence for all algebras A.

A second (but simpler for us) problem concerns a property and question of Pereira related to free subsets, internally approachable models, and the PCF conjecture.

Covering Lemma arguments using inner models provide lower bounds for both properties.

2008-11-20Antonín Kučera (Charles University, Prague)Algorithmic randomness (thick Π01 classes)
In the first part of the talk both a survey of various approaches to algorithmic randomness (including Martin-Löf tests, prefix-free Kolmogorov complexity, martingales) and basic facts about properties of 1-random sets will be given. The role of Π01 classes of positive measure will be pointed out (as a kind of thick Π01 classes). Generalizations and modifications of the concept of 1-randomness will be mentioned.

The second part of the talk deals with algorithmic weakness, namely with the class of K-trivials and its various other characterizations such as lowness for randomness etc. Among other things an existence of a low T-upper bound for the class of K-trivials will be presented as a corollary of a more general characterization of ideals in T-degrees which have a low T-upper bound (a result of Kucera-Slaman). A substantial role of PA sets (i.e. {0, 1}-valued DNR functions), which form another kind of thick Π01 classes, will be explained. If time permits, LR-reducibility (LR stands for low for random) will be explained together with its importance.
2008-11-27Vera Fischer (KGRC)Further combinatorial properties of Cohen forcing
Relaying on some combinatorial properties of Cohen forcing, we suggest a generalization of Mathias forcing which preserves the unboundedness of small unbounded families. The construction can be considered a first step towards obtaining the consistency of \(\omega_1< \mathfrak{b}<\mathfrak{b}^+<\mathfrak{s}\).

2008-12-04Luca Motto Ros (KGRC)Analytic Equivalence Relations and Bi-Embeddability
The analysis of the structure of analytic equivalence relations (ER for short) under Borel-reducibility has been one of the most relevant subject in the recent history of Descriptive Set Theory. Among ERs, a special place is occupied by the isomorphism relation on countable models of a certain Lω1-sentence φ. However, isomorphism relations are just a special case, as there are many example of ERs which are even not Borel-reducible to an isomorphism relation. On the contrary, we will show in this talk that the bi-embeddability (resp. bi-homomorphism, bi-weak-homomorphism) relation is able to capture the whole complexity of the ER-structure: for every ER E there is an Lω1-sentence φ such that E is Borel-equivalent to bi-embeddability (resp. bi-homomorphism, bi-weak-homomorphism) on the collection of countable models of φ. This is joint work with Sy D. Friedman.

2008-12-11Asger Törnquist (KGRC)Automorphisms of S_infty/fin, part II
2008-12-17Grzegorz Plebanek (University of Wroclaw)On Boolean algebras and C(K) spaces
2008-12-18Mirna Dzamonja (University of East Anglia)The confluence between model theory and set theory in Banach spaces
We shall present some recent results where ideas from mathematical logic have been proved useful in solving probems in Banach spaces. We shall especially concentrate on the emerging possibilities of solving model-theoretic methods to solve concrete problems in Banach space theory.