Mathematical Logic: Set theory and model theory
2009 fall
Seminar program
The seminars will be held on Wednesdays and Thursdays during the semester.
Thursday, December 10, 2009:
14.00-15.00 Stevo Todorcevic, University of Toronto
A basis problem for compact spaces
Abstract: Using Forcing Axioms we analyse a possible structure of the class of
compact spaces. We also look at formulations of the problem in some of
its dual or double-dual forms in classes of Boolean algebras, Banach
spaces and Choquet simplices.
15.30-16.30 Jouko Väänänen, Universitites of Helsinki and of Amsterdam
Set theory and logic
Abstract: I will discuss the relationship between two ways to investigate mathematical structures, namely the set theoretic way and the model theoretic way. I compare hierarchies of both and establish level by level matches. I will discuss ``sort logic" as the ultimate limit of model theoretic languages.
Wednesday, December 9, 2009:
14.00-15.00 Erik Palmgren, Uppsala University
Constructivist and structuralist foundations: Bishop's and Lawvere's theories of sets
Abstract: Bishop's informal set theory is briefly discussed and compared to Lawvere's Elementary Theory of the Category of Sets (ETCS). We then present a constructive and predicative version of ETCS, whose standard model is based on the constructive type theory of Martin-Löf. The theory, CETCS, provides a structuralist foundation for constructive mathematics in the style of Bishop.
15.30-16.30 Bill Mitchell, University of Florida, Gainesville
The covering lemma: 35 years, and a question
Abstract:This talk will survey the development of the covering lemma, and associated core models, with an emphasis on the two breakthroughs of Jensen's original discovery and the intervention of Woodin cardinals. At the end I will observe that in the presence of a Woodin cardinal the covering lemma dwindles to almost nothing, and ask whether any more can be saved.
14.00-15.00 Stevo Todorcevic, University of Toronto
A basis problem for compact spaces
Abstract: Using Forcing Axioms we analyse a possible structure of the class of
compact spaces. We also look at formulations of the problem in some of
its dual or double-dual forms in classes of Boolean algebras, Banach
spaces and Choquet simplices.
15.30-16.30 Jouko Väänänen, Universitites of Helsinki and of Amsterdam
Set theory and logic
Abstract: I will discuss the relationship between two ways to investigate mathematical structures, namely the set theoretic way and the model theoretic way. I compare hierarchies of both and establish level by level matches. I will discuss ``sort logic" as the ultimate limit of model theoretic languages.
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14.00-15.00 Stevo Todorcevic, University of Toronto
A basis problem for compact spaces
Abstract: Using Forcing Axioms we analyse a possible structure of the class of
compact spaces. We also look at formulations of the problem in some of
its dual or double-dual forms in classes of Boolean algebras, Banach
spaces and Choquet simplices.
15.30-16.30 Jouko Väänänen, Universitites of Helsinki and of Amsterdam
Set theory and logic
Abstract: I will discuss the relationship between two ways to investigate mathematical structures, namely the set theoretic way and the model theoretic way. I compare hierarchies of both and establish level by level matches. I will discuss ``sort logic" as the ultimate limit of model theoretic languages.
Thursday, December 10, 2009:
14.00-15.00 Stevo Todorcevic, University of Toronto
A basis problem for compact spaces
Abstract: Using Forcing Axioms we analyse a possible structure of the class of
compact spaces. We also look at formulations of the problem in some of
its dual or double-dual forms in classes of Boolean algebras, Banach
spaces and Choquet simplices.
15.30-16.30 Jouko Väänänen, Universitites of Helsinki and of Amsterdam
Set theory and logic
Abstract: I will discuss the relationship between two ways to investigate mathematical structures, namely the set theoretic way and the model theoretic way. I compare hierarchies of both and establish level by level matches. I will discuss ``sort logic" as the ultimate limit of model theoretic languages.
Abstract: Using Forcing Axioms we analyse a possible structure of the class of
compact spaces. We also look at formulations of the problem in some of
its dual or double-dual forms in classes of Boolean algebras, Banach
spaces and Choquet simplices.
15.30-16.30 Jouko Väänänen, Universitites of Helsinki and of Amsterdam
Set theory and logic
Abstract: I will discuss the relationship between two ways to investigate mathematical structures, namely the set theoretic way and the model theoretic way. I compare hierarchies of both and establish level by level matches. I will discuss ``sort logic" as the ultimate limit of model theoretic languages.
Thursday, December 3, 2009:
14.00-15.00 Heike Mildenberger, Universität Wien
Filters versus semifilters
Abstract:
The filter dichotomy principle says: For every non-meagre filter
there is a finite-to-one function mapping it to an ultrafilter.
The semifilter trichotomy principle says: For every non-meagre
upwards closed subset of the set of infinite subsets of the natural
numbers there is a finite-to-one function mapping it to an ultrafilter
or mapping it to the set of all infinite subsets of the
natural numbers.
Blass and Laflamme showed: In the Blass-Shelah model and in all other models of
(u g) the semifilter trichotomy holds, and the trichotomy is indeed equivalent to (u g).
Recently I found a model of the filter dichotomy in which the
semifilter trichotomy does not hold. This answers Blass' 1989
question whether (u g) is strictly stronger than the filter
dichotomy affirmatively. In the talk we will look at some of these
rather combinatorial forcing steps.
15.30-16.30 James Cummings, Carnegie Mellon University, Pittsburgh
Recent uses of infinitary methods in finite combinatorics
Abstract:
Over the past few years there have been some striking new
applications of infinitary methods in finite combinatorics. I'll
describe one of these methods, Razborov's "flag algebras". This
is primarily a survey talk.
Wednesday, November 25, 2009:
14.00-15.00 Jean Larson, Gainesville, Florida
Scattered thoughts on infinite combinatorics
Abstract: Two themes in the history of infinite combinatorics in the twentieth century will be explored: the partition calculus and order, especially trees. The partition calculus begins with early results like Ramsey's theorem on the existence of infinite homogeneous sets for finite partitions of the r-element subsets of an infinite set. The talk will detail some of the extensions of this result into the uncountable, as well as looking at a variety of examples that delineate the limits to what can be achieved. The study of order begins with Hausdorff's investigations into generating sets, bases and types universal for various collections of order types. Add the Suslin problem which was translated by Kurepa into a question about trees to get a rich collection of questions which have been studied with increasingly sophisticated techniques over the twentieth century.
15.30-16.30 Hajime Ishihara, Japan Advanced Institute of Science and Technology, Ishikawa
A boundedness principle in constructive reverse mathematics
Abstract:
A subset S of N is bounded if there exists K such that s < K for all s in S; pseudobounded if lim s_n / n = 0 for each sequence (s_n) in S. It is easy to see that if S is bounded, then it is pseudobounded. We will deal with the following boundedness principle in the context of constructive reverse mathematics.
BD-N. Each countable pseudobounded subset of N is bounded.
The boundedness principle BD-N is provable in the weak classical
system RCA_0, derivable from an intuitionistic principle -- a
version of Brouwer's continuity principle (WC-N), and derivable from
the principles of constructive recursive mathematics, that is,
Extended Church's thesis (ECT_0) and Markov's principle (MP).
However, Peter Lietz (2004) showed that BD-N is not provable in
E-HA^\omega + AC -- a natural formal system for Bishop's
constructive mathematics. We will give a survey of the boundedness principle BD-N, its
equivalents and the recent results in constructive reverse
mathematics.
Thursday, November 19, 2009:
14.00-15.00 Vera Koponen, University of Uppsala
Asymptotic probabilities of extension axioms.
Abstract:
Consider a class K of finite structures, in
a finite relational language. We investigate
when it is the case that, for every extension
axiom Ext, compatible with K, the probability
that Ext is true in a member M of K approaches 1
as the cardinality of M tends to infinity; and when,
for at least one extension axiom, the probability
of it being true remains below some c < 1
(possibly approaching 0). First we do this
for the uniform probability measure and then
for the 'dimension conditional measure', which
is more "generous" with respect to satisfiability
of extension axioms. From the results obtained we
can derive zero-one laws and we get some information
about when the Fraisse limit of K, if it exists,
is a model of the 'almost sure theory' of K.
15.30-16.30 Kerkko Luosto, University of Helsinki
Combinatorics of quantifiers
Abstract: The motivation to study generalized quantifiers is mostly applicative:
quantifiers are commonly used tools in theoretical computer science
and linguistics. Here, I like to take another point a view, taking a
look at quantifier definability problems and related mathematics. To
tackle these problems, one needs mostly combinatorial tools.
Sometimes pigeonhole principle and elementary counting suffice, but
usually one needs something stronger, e.g., Ramsey theory or some kind
of combinatorics of words. In this vein, I shall survey some of the
results of the field
Wednesday, November 18, 2009:
14.00-15.00 Joan Bagaria, Universitat de Barcelona
Large cardinals and accessible categories
Abstract: Many natural questions in category theory depend on set theory. A typical example is whether every full limit-closed subcategory of a given complete category is reflective. While there are counterexamples involving the category of topological spaces and continuous functions, if the given category is locally-presentable, then the question is equivalent to the large cardinal notion known as Vopenka's Principle (VP). We shall present an analysis of the large cardinal hierarchy ranging from supercompact cardinals to VP which can be used to show, for example, that for definable full limit-closed subcategories of locally-presentable categories, reflexivity follows from much weaker large-cardinal assumptions than VP.
15.30-16.30 Philipp Schlicht, Friedrich-Wilhelm-Universität, Bonn
Descriptive set theory at uncountable cardinals
Abstract:I would like to give an introduction to analogues of classical results in descriptive set theory for the space kappa to the kappa. I will mention the perfect set property and the property of Baire and describe what is different from the countable case.
Thursday, November 12, 2009:
14.00-15.00 Lauri Hella, University of Tampere
Monadic sigma-1-1 and modal logic with quantified binary relations
Abstract: We investigate the expressive power of a range of modal logics extended with
second-order prenex quantification of binary and unary relations. Our main result is that
Sigma-1-1(BML=), i.e., Boolean modal logic extended with identity modality and
existential prenex quantification of binary and unary relations, translates to monadic
Sigma-1-1. As a corollary, we get decidability results for multimodal logics
on various classes of Kripke frames. Our result can also be seen as a step towards
showing that Sigma-1-1(FO2) is contained in monadic Sigma-1-1, i.e., each existential
second-order sentence with first-order part containing only two variables is equivalent
to a sentence in monadic Sigma-1-1. This was conjectured by Grädel and Rosen in 1999.
Joint work with Antti Kuusisto (University of Tampere)
15.30-16.30 Juha Kontinen, University of Helsinki
Regular representations of uniform TC0.
Abstract: The circuit complexity class DLOGTIME-uniform AC0 is known to be a modest subclass of DLOGTIME-uniform TC0. The weakness of AC0 is caused by the fact that AC0 is not closed under restricting AC0-computable queries into AC0-computable substrings of the input. Analogously, in descriptive complexity, the logics corresponding to DLOGTIME-uniform AC0 do not have the relativization property and hence they are not regular. This weakness of DLOGTIME-uniform AC0 has been elaborated in the line of research on the Crane Beach Conjecture. The conjecture (which was refuted by Barrington, Immerman, Lautemann and Schweikardt in 2001) was that if a language L has a neutral letter, then L can be defined in FOB, first-order logic with the collection of all numeric built-in predicates B, iff L can be already defined in FO{<}, first-order logic with order. We consider logics in the range of DLOGTIME-uniform AC0 and TC0. First we show that DLOGTIME-uniform TC0 can be logically characterized in terms of quantifier logics with cardinality quantifiers FO{<}(C_S), where S is the range of some polynomial with positive integer coefficients of degree at least two. In the second part of the paper we first adapt the key properties of general logics to accomodate built-in relations. Then we define the regular interior R-int(L) and regular closure R-cl(L), of a logic L, and show that the Crane Beach Conjecture can be interpreted as a statement concerning R- int(FO_B). In particular, by the result of Barrington, Immerman, Lautemann and Schweikardt, if B contains only unary relations (besides <) then R-int(FO_B)=FO_{<} on strings. In contrast, if B contains < and the range of a polynomial of degree at least two, then R-cl(FO_B) includes all languages in DLOGTIME-uniform TC0. Joint work with Lauri Hella and Kerkko Luosto.
Thursday, November 5, 2009:
14.00-15.00 Agatha Walczak-Typke, University of Vienna
Constructibility of potentially isomorphic pairs vs
classification in homogeneous model theory.
Abstract: The work presented is joint with S-D Friedman and T
Hyttinen. Our aim was to generalize a very nice result of Friedman,
Hyttinen, and Rautila which tied first-order model theoretic
classification theory to constructibility under the assumption of 0-
sharp, to a non-elementary model theoretic setting. The original
result stated:
Theorem. Assume 0-sharp 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-one in L. Then the
collection of constructible pairs of models A,B of T, of size 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 because of the already well developed
structure/non-structure theory in this setting. I will present the
result for Homogeneous Model Theory analogous to the theorem above,
and discuss some of the issues involved in the proof.
15.30-16.30 Carlos Di Prisco, IVIC, Caracas
Graphs defined by the shift operation
Abstract: We consider graphs defined by the shift on spaces like the
Baire space or
the Cantor space, and prove several results regarding their Borel
chromatic numbers.
Thursday, October 29, 2009:
14.00-15.00 Ralf Schindler, Universität Münster
Woodin's axiom (*), bounded forcing axioms, and related issues. Part II
Abstract:
The P-max extension of L(R) does not satisfy BMM, and in fact
under (*), BPFA is equivalent to BMM++. The forcing which verifies
natural Pi-2 statements under BMM + "NS is precipitous" is semi-proper
if and only if all stationary set preserving forcings are semi--proper.
15.30-16.30 Daisuke Ikegami, Universiteit van Amsterdam
Blackwell determinacy
Abstract:
Blackwell games are infinite games with imperfect information which originally come from game theory while Gale-Stewart games are infinite games with perfect information. Blackwell determinacy was introduced as an extension of von Neumann's minimax theorem and has been investigated by several people in set theory and game theory but much less than the determinacy of Gale-Stewart games. In 1998, Martin proved that the Axiom of Determinacy (AD) implies the Axiom of Blackwell Determinacy (Bl-AD) and conjectured the converse, which is still not known to be true. In this talk, I will introduce Blackwell games and Blackwell determinacy, give some ideas of the proofs of the consequences of Bl-AD and the Axiom of Real Blackwell Determinacy (Bl-AD_R), and discuss the problem how far one can be close to the Axiom of Real Determinacy (AD_R) from Bl-AD_R.
Thursday, October 22, 2009:
14.00-15.00 Meeri Kesälä, University of Helsinki
Finitary abstract elementary classes
Abstract: We discuss types and independence in a non-elementary context.
15.30-16.30 Fredrik Engström, University of Gothenburg
Logical constants and invariance
Abstract: According to Tarski an operation on a domain should be
counted as a logical constant iff it is invariant under all
permutations of the domain, thus defining logic as the study of
permutation invariant operations. Tarski's thesis has been heavily
criticized (on good grounds) for generating way to many logical
constants. In the last twenty years there have been some alternative
suggestions for characterizing logical constancy in terms of invariant
operators. I will try to give an introduction to this field and also
prove some new results.
Wednesday, October 21, 2009:
14.00-15.00 Menachem Magidor, the Hebrew University, Jerusalem
The Lowenheim-Skolem-Tarski number of the Hartig-quantifier can be the first inaccessible
Abstract: In this talk I sketch the proof of the following theorem: If ZFC + "There is a supercompact cardinal" is consistent, then so is ZFC + "There is an inaccessible cardinal" + "The downward Lowenheim-Skolem-Tarski theorem for the Hartig quantifier holds for the first inaccessible cardinal". The assumption of a supercompact cardinal seems almost unavoidable, since this Lowenheim-Skolem-Tarski theorem implies the failure of weak square for large enough cardinals. (Joint work with Jouko Väänänen.)
15.30-16.30 Benno van den Berg, TU Darmstadt
An introduction to algebraic set theory
Abstract: In this talk I will give an introduction to "algebraic set
theory". Algebraic set theory is an approach to the semantics of set
theory based on categorical logic and inspired by topos theory. I will
concentrate on the main ideas, rather than on technical results.
Thursday, October 15, 2009:
14.00-15.00 Tapani Hyttinen, University of Helsinki
Model theory and metric structures
Abstract:In metric structures there are other natural notions
of isomorphism than the one used in model theory.
I will discuss how these look like from the point of
view of model theory.
15.30-16.30 Vadim Kulikov. University of Helsinki
Equivalence relations on a generalized Baire space
Abstract: Our aim is to study similar questions that are studied in
descriptive set theory but in the setting, where the Baire space is
replaced by the space of functions from an uncountable cardinal to
itself. Specially we are interested in the complexity of isomorphism
relations of structures of given theories. We also study Borel
reducibility between isomorphism relations and set theoretical
equivalence relations such as NS.
Wednesday, October 14, 2009:
14.00-15.00 Boban Velickovic, Université Paris 7
Universal countable Borel equivalence relations
Abstract: We survey some recent results on countable Borel equivalence
relations with the particular emphasis on techniques used
to show that a given equivalence relation is universal among
all countable Borel equivalence relations.
15.30-16.30 Andrés Villaveces, Universidad Nacional de Colombia, Bogota
Dependence outside first order contexts
Abstract: I will explore connections between First Order NIP
(=dependent) theories, Shelah's Generic Pairs Conjecture, and (new)
non-elementary versions of dependence. In particular, I will describe
recent joint work with Grossberg and VanDieren on a new independence
notion (splintering), provide examples, and (time-permitting) an
application to a problem in AEC.
Thursday, October 8, 2009:
14.00-15.00 Dag Normann, University of Oslo
The sequential functionals are far from being algebraic domains
Abstract: In 2005 I proved that the sequential functionals of pure type 3 is not a
dcpo, and thus does not coincide with the interpretation of pure type 3 in Milner's fully abstract model for PCF.
In two papers from 2007 and 2008, V. Sazonov formulated four conjectures, all expressing that the sequential functionals in general form different kinds of partial orderings than those studied in domain theory.
In this talk, I will report from a research project jointly with Sazonov, where his conjectures are verified in a strong sense.
We will characterize the set of types for which the sequential functionals form a dcpo, and show that sequential functionals in general do not commute with least upper bounds of directed sets.
We will give a brief introduction to the intentional finite sequential
procedures leading to the extensional finite sequential functionals, prove one or two technical results and survey other results from this joint project.
15.30-16.30 Jaap van Oosten, University of Utrecht
Constructions of (order-) partial combinatory algebras
Abstract: By looking at sequential functionals on spaces of functions, we can generalize Kleene's pca of functions $N^N$ to sets of functions $A^A$ for arbitrary infinite A. Substructures will be considered, and universal properties w.r.t. Longley's category of pcas and decidable applicative morphisms.
Thursday, October 1, 2009:
14.00-15.00 Hugh Woodin, University of California, Berkeley
Pi-2-2 -maximality and generic diamond
Abstract:"Generic diamond" is the assertion that the theory of
H(omega-2) is absolute between V and V[G] where G is a V-generic subset
of omega-1. Suppose phi and psi are Pi-2-2-sentences each of which can
hold is some generic extension of V in which generic diamond holds. Must
phi and psi be mutually consistent with generic diamond? The
corresponding question for CH has been answered negatively by
Aspero-Larson-Moore. Suppose the answer to this question is also
negative. Then M-infty does not exist where M-infty is the "largest"
Mitchell-Steel mouse. The same result is true with Pi-2-2 replaced by
Pi-2-2(NS) where NS is the nonstationary ideal on omega-1.
15.30-16.30 Ralf Schindler, Universität Münster
Woodin's axiom (*), bounded forcing axioms, and related issues.
Abstract: :The P-max extension of L(R) does not satisfy BMM, and in fact
under (*), BPFA is equivalent to BMM++. The forcing which verifies
natural Pi-2 statements under BMM + "NS is precipitous" is semi-proper
if and only if all stationary set preserving forcings are semi--proper.
Wednesday, September 30, 2009:
14.00-15.00 Jana Flaskova, University of West Bohemia, Plzeń
Some ultrafilters on natural numbers
Abstract: The background for this talk form two concepts relating
ultrafilters on natural numbers with "small" subsets of natural numbers:
I-ultrafilters and 0-points. We will construct an ultrafilter whose
properties match to some extent both concepts, namely an ultrafilter u
such that for every one-to-one function f there is a set U in u such
that f[U] belongs to the summable ideal. The construction relies on
elementary knowledge of calculus and elaborated combinatorics.
15.30-16.30 Thomas Streicher, Technische Universität Darmstadt
Sheaf models for CZF refuting power set and full separation
Abstract: Categorical Logic, e.g. sheaf toposes, give rise to a huge
variety of models for IZF often validating interesting axioms
incompatible with classical logic. IZF is known to be equiconsistent
with ZF and thus fairly strong. The theory CZF as introduced by P.Aczel
(and J.Myhill) is a much weaker system equiconsistent with Martin-Löf
type theory. It lacks the powerset axiom and separation is restricted to
bounded formulas but one has function sets as ensured by the Fullness
Axiom. We show that in certain sheaf models restricting the powerset
operation one can construct models of CZF which refute both the powerset
axiom and the full separation scheme. (Joint work with A. Simpson, Univ.
Edinburgh.)
Thursday, September 24, 2009:
14.00-15.00 Roman Kossak, City University of New York
Classification problems in models of Peano arithmetic
Abstract: We examine how certain classification problems for countable models of PA fit into the framework of Borel reductions of equivalence relations on standard Borel spaces. This is joint work with Samuel Coskey.
15.30-16.30 Marcin Sabok, Wroclaw University
Playing with idealized forcing
Abstract: I will begin with an elementary introduction to the idealized forcing. Then I will show some recent contributions to the theory. As an application, I will show some infinite-dimensional "perfect-set theorems".
Thursday, September 17, 2009:
14.00-15.00 Paul Larson, Miami University, Ohio
Universally measurable sets in generic extensions
Abstract: A subset of a Polish space is said to be universally measurable if it is measured by every complete Borel measure on the space.
In a forcing extension by a suitably-sized random algebra, there are just continuum many universally measurable sets (this result is
joint with Shelah and answers a question of Mauldin), and the universally measurable sets can be characterized in terms of unions of
Borel sets (joint with Neeman and Shelah). We will also discuss the consistent non-existence of universally measurable measures on
the integers.
15.30-16.30 Justin Moore, Cornell University, Ithaca
CH and the combinatorics of the club filter
Abstract: I will discuss open problems and recent work concerning combinatorial properties of the club filter in the context of the Continuum Hypothesis.
Wednesday, September 16, 2009:
14.00-15.00 Saharon Shelah, Hebrew University and Rutgers University
An advance on the existence of completely separable MAD families
Abstract: We throw some light on the question: is there a MAD family (= a maximal family of infinite subsets of N, the intersection of any two is finite) which is completely separable (i.e. any subset X of N is included in a finite union of members of the family *or* includes a member (and even continuum many members) of the family). We prove that it is hard to prove the consistency of the negation (a) If the continuum is below alef-omega, then there is such a family. (b) If there is no such families then some situation related to pcf holds whose consistency is large; and if frak-a is greater than aleph-1 even unknown. (c) If there is no inner model with measurables, then there is such a family.
15.30-16.30 Ali Enayat, American University, Washington DC
Automorphisms of Models of Set Theory
Abstract: : I will provide a survey of various results (some old, but mostly new) on automorphisms of models of set theory, with an eye towards:
(a) comparisons/contrasts of the behaviour of automorphisms of models of PA and models of ZFC; and
(b) applications to the Quine-Jensen system NFU of set theory with a universal class
Thursday, September 10, 2009:
14.00-15.00 Alexander Kechris, CALTECH, Pasadena
The complexity of classification problems in ergodic theory.
Abstract: The last two decades have seen the emergence of a theory of set theoretic complexity of classification problems in mathematics. In this talk, I will survey recent developments concerning the application of this theory to classification problems in ergodic theory.
15.30-16.30 Istvan Juhasz, Hungarian Academy of Sciences, Budapest
On the convergence and character spectra of compact spaces
Abstract: An infinite set A in a space X converges to a point p (denoted by A --> p) if for every neighborhood U of p we have |A\U| < |A|. We call cS(p,X) = { |A| : A subset X and A --> p } the convergence spectrum of p in X and cS(X) = cup{cS(x,X) : x in X} the convergence spectrum of X. The character spectrum of a point p in X is chi-S(p,X) = {chi(p,Y) : p is non-isolated in Y subset X} and chi-S(X) = cup{chi-S(x,X) : x in X} is the character spectrum of X. If kappa in chi-S(p,X) for a compactum X then {kappa,cf(kappa)} subset cS(p,X). A selection of our joint results with W. Weiss (X is always a compactum): (1) If X is countably tight then chi(p,X) > lambda =(lambda to the power omega) implies lambda is in chi-S(p,X). (2) If chi(X) > (2 to the power kappa), then kappa+ is in cS(X), in fact there is a converging discrete set of size kappa+ in X. (3) If we add lambda Cohen reals to a model of GCH then in the extension for every kappa less than or equal to lambda, there is X with chi-S(X) = {omega,kappa}. (4) If all members of chi-S(X) are limit cardinals then |X| is less than or equal to (sup{|(closure of S)| : S in ([X] to the power omega)}) to the power omega: (5) It is consistent that (2 to the power omega) is as big as you wish and there are arbitrarily large X with chi-S(X) cap (omega,(2 to omega)) = emptyset.
Wednesday, September 9, 2009:
14.00-15.00 John Baldwin, University of Illinois, Chicago
Shelah´s Conjecture: The Universe is Wide or Deep
15.30-16.30 Ilijas Farah, York University, Toronto
Model theory of operator algebras (joint work with Bradd Hart and David Sherman)
Abstract: McDuff and Kirchberg have considered whether the ultrapower and the relative commutant of a II$_1$ factor or a C*-algebra, respectively, depend on the choice of the ultrafilter. I will show that the negative answer to each of these questions is equivalent to the Continuum Hypothesis, extending results of Ge--Hadwin and myself. I will also outline a version of `model theory for metric structures' suitable for study of C*-algebras and tracial von Neumann algebras.
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