2013-01-07Aleksander Ivanov and Barbara Majcher-Iwanow (University of Wrocław, Poland)Polish G-spaces similar to logic G-spaces of continuous structures

We extend the concept of nice topologies of H.Becker to the general case of Polish \(G\)-spaces (Becker assumed that \(G\lt Sym(\omega)\)). Our apprach is based on continuous first order logic.

Let \(({\bf Y},d)\) be a Polish space and \(Iso({\bf Y},d)\) be the corresponding isometry group endowed with the pointwise convergence topology. Then \(Iso ({\bf Y},d)\) is a Polish group. It is worth noting that any Polish group \(G\) can be realised as a closed subgroup of the isometry group \(Iso ({\bf Y},d)\) of an appropriate Polish space \({\bf Y}\).

For any countable continuous signature \(L\) the set \({\bf Y}_L\) of all continuous metric \(L\)-structures on \(({\bf Y},d)\) can be considered as a Polish \(Iso({\bf Y},d)\)-space. We call this action {\em logic}. Note that for any tuple \(\bar{s}\in {\bf Y}\) the map \(g\rightarrow d(\bar{s},g(\bar{s}))\) can be considered as a graded subgroup of \(Iso({\bf Y},d)\). For any continuous sentence \(\phi\) we have a graded subset of \({\bf Y}_L\) defined by \(M \rightarrow \phi^{M}\).

We investigate Polish \(G\)-spaces \({\bf X}\) where \(G\) is Polish. We pove that distinguishing an appropriate family of graded subgroups of \(G\) and some family \(\mathcal{B}\) of graded subsets of \({\bf X}\) (called a graded nice basis) we arrive at the situation very similar to the logic space \(\mathcal{U}_L\), where \(\mathcal{U}\) is the bounded Urysohn space. Treating elements of \(\mathcal{B}\) as continuos formulas we obtain topological generalisations of several theorems from logic, for example Ryll-Nardzewski theorem.

2013-01-10Peter Holy (Bristol University, UK)Large Cardinals and Lightface Definable Wellorders without GCH
We investigate the possible coexistence of wellorders of \(H_{\kappa^+}\) which are lightface definable over \(H_{\kappa^+}\) for inaccessible cardinals \(\kappa\) with a failure of the GCH at \(\kappa\). Assuming SCH it is possible to obtain a lightface definable wellorder of \(H_{\kappa^+}\) for every inaccessible \(\kappa\) while preserving all cofinalities, the continuum function and all supercompact cardinals. This is joint work with Sy Friedman and Philipp Lücke, who jointly obtained a similar result for boldface (instead of lightface) definable wellorders.
2013-01-17Jana Flašková (University of West Bohemia in Pilsen, Czech Republic)Van der Waerden ideal and its cardinal invariants
The sets which do not contain arbitrarily long arithmetic progressions form an ideal on the natural numbers, which is called van der Waerden ideal. The structure of the van der Waerden ideal and its relation to other well-known ideals on the natural numbers will be the subject of this talk. We shall also discuss some cardinal invariants of the ideal such as additivity, cofinality, uniformity and covering number. For example, we will show that the uniformity number of the van der Waerden ideal is less or equal to the reaping number.
2013-01-24Claudio Ternullo (KGRC)Does set theory refute Platonism?
Platonism is an influential and well-established conception of mathematics and mathematical practice, which states that the axioms of mathematics ''refer'' to an independently existing realm of mathematical entities.

In particular, set-theoretic platonism entails the view that the axioms of set theory refer to an independently existing realm of objects (the ''universe of sets'') and that, given any reasonably formulated set-theoretic statement φ, φ has a ''definite'' truth-value.

In my talk, I will address the question of whether there any sufficient grounds to advocate platonism in view of contemporary set theory. In particular, the ''truth-value indeterminacy'' connected to the independence phenomenon in ZFC and its extensions does not seem to validate platonists’ epistemic optimism concerning our ability to have access to ''unique'' truths.

I will be particularly concerned with Gödel’s platonism and its ontological and epistemological implications, but I also intend to pay attention to several alternative formulations which are deemed to belong to the big family of realism in mathematics. Naturalism, full-blooded platonism, extreme platonism, conceptual realism will also be taken into account. My aim is to show that they may not perform better than Gödel’s conception.

2013-01-31Massoud Pourmahdian (IPM, Tehran, Iran)First order Goedel logics
In this talk I will talk on different notions of compactness in first order Goedel logic. Then I consider an extension of first order Goedel logic by adding the rational numbers as logical constants. Some model theory is developed for this logic and in particular the Robinson consistency theorem is shown for this extension. I also give some general comments as how to define the metric Goedel logic similar to metric (Lukasewicz) logic.

2013-03-07Sy-David Friedman (KGRC)Vaught's Conjecture, the Generic Morley Tree and Fragment Embeddings
If \(\varphi\) is a scattered sentence of \(L_{\omega_1\omega}\) (i.e., one with at least one model but no perfect set of countable models) then associated to \(\varphi\) is its Morley tree. Each node of this tree is a countable theory which is atomic for a countable fragment of \(L_{\omega_1\omega}\) containing \(\varphi\). The Morley tree has height at most \(\omega_1\) and Vaught's conjecture asserts that its height is in fact less than \(\omega_1\). Using a generic version of the Morley tree together with a notion of fragment embedding, I'll prove a theorem of Harrington which states that if \(\varphi\) is a counterexample to Vaught's conjecture then there are models of \(\varphi\) with Scott rank arbitrarily large below \(\omega_2\).

2013-03-14Moritz Müller (KGRC)Weak pigeonhole principles
The pigeonhole principle states that, if \(n+1\) pigeons fly into \(n\) holes then some hole must be doubly occupied. Weak pigeonhole principles state the same but for \(2n\) or \(n^2\) many pigeons. The talk surveys results and open problems concerning the proof complexity of these principles. Then some new results concerning a relativized such principle are presented. This principle states that if at least \(2n\) out of \(n^2\) many pigeons fly into \(n\) holes, then some hole must be doubly occupied. This is joint work with Albert Atserias and Sergi Oliva.
2013-03-21Yurii Khomskii (KGRC)Cichon's diagram and regularity properties
The study of regularity properties in descriptive set theory is closely related to cardinal invariants of the continuum. Specifically, if we consider the Baire property, Lebesgue measure, and the Laver-, Miller- and Sacks-regularity for \(\Sigma^1_2\) and \(\Delta^1_2\) sets of reals, we obtain a pattern analogous to the well-known Cichon's diagram.

In this project we want to see what happens to this diagram on the \(\Sigma^1_3\), \(\Delta^1_3\) and higher projective levels. It is well-known that in the presence of a measurable cardinal, the picture lifts from the second to the third level, but without these assumptions, one can observe more interesting and surprising patterns. Consequently, all our results are proved on the basis of ZFC alone or ZFC with an inaccessible cardinal. An important technical ingredient in our proofs is the use of Suslin and Suslin+ proper (not necessarily ccc) forcing.

This is joint work with Vera Fischer and Sy Friedman.

2013-04-11Martin Koerwien (KGRC)Around characterizing aleph_1
An \(L_{\omega_1,\omega}\) sentence characterizes a cardinal \(\kappa\) if it has a model of size \(\kappa\) but no model in \(\kappa^+\). We study the known examples of complete sentences that characterize \(\aleph_1\) and observe several notable phenomena about them. Our goal is to understand the mechanisms that make a sentence characterize \(\aleph_1\). This is related to some recent developments:

(1) Hjorth showed that if there is a counterexample to Vaught's conjecture, there is also one that characterizes \(\aleph_1\). So it is tempting to try proving Vaught's conjecture by showing that every counterexample must have a model in \(\aleph_2\) (which moreover a result of Harrington's suggests). This however turns out to be a red herring.

(2) While we know the notion of a complete sentence having a model in kappa is absolute for \(\kappa=\aleph_1\) and non-absolute for \(\kappa=\aleph_3\), even assuming GCH, this is still an open issue for \(\kappa=\aleph_2\).

2013-04-18Hubie Chen (Universidad del País Vasco, Spain and Basque Foundation for Science, Spain)Model Checking Quantified-Conjunctive Formulas
Model Checking, the problem of deciding if a logical sentence holds on a structure, is a basic computational task which is well-known to be computationally intractable in general. We study model checking on the quantified-conjunctive fragment of first-order logic, by which is meant the class of formulas built using conjunction as the only connective, and both quantifiers. We present complexity classification results which systematically identify tractable cases of model checking on this fragment. One recent such classification result involves an algebraic understanding of logical equivalence in this fragment, which understanding we discuss.
2013-04-25Valentin Bura (TU Wien, Austria)Reverse Mathematics of Divisibility in Integral Domains
After a brief introduction to the research program of Reverse Mathematics, we consider the statement "If an integral domain satisfies the ACCP, it is Atomic". We give an outline of our proof that this Theorem is equivalent to ACA0 over RCA0. We conclude by discussing a related question that we left open.

This is joint work with Noam Greenberg and Dan Turetski.
2013-05-02Radek Honzik (KGRC)More on the tree property
More than a Woodin cardinal is required to obtain tree property at two adjacent cardinals. We make a first step toward showing that a much weaker assumption is sufficient for the tree property to hold at all even successor cardinals. Specifically, we show that from \(\omega\)-many weak compacts one can obtain the tree property at all of the \(\aleph_{2n}\)'s and from a cardinal which is strong up to a larger weak compact one can in addition have the tree property at \(\aleph_{\omega+2}\) (\(\aleph_\omega\) strong limit).

This work is joint with Sy-D. Friedman.

2013-05-16Philipp Schlicht (University of Bonn, Germany)On Generalized Choquet Spaces and Groups
We introduce an analogue to Polish spaces for uncountable regular cardinals \(\kappa\) with \(\kappa^{\lt\kappa} = \kappa\) via a variant of the Choquet game of length \(\kappa\). There is a surjectively universal such space, and any two such spaces of size \(> \kappa\) with no points which are the intersection of fewer than \(\kappa\) open sets are \(\kappa\)-Borel isomorphic. We consider the special case of generalized \(\kappa\)-valued ultrametric spaces with the property that the intersection of any decreasing sequence of balls is nonempty and construct a family of universal Urysohn spaces. We then prove that the logic action of Sym(\(\kappa\)) is universal for \(\kappa\)-Borel measurable actions of Sym(\(\kappa\)) with respect to equivariant embeddings.

This is joint work with Samuel Coskey.

2013-06-06Yann Strozecki (Versailles University, France)Simple stochastic games: a state of the art
In this talk I present simple stochastic games and some related games as parity or discounted payoff games. They can be used to solve interesting problems, such as the model checking of the mu calculus. Moreover, these games generalize Markov decision processes, repeated games, and linear programming which is perhaps the most ubiquituous technique in computer science.

The main problem is to compute optimal strategies for the two players, which seems to be hard because of the randomness and because a play can be infinite. It turns out that the problem is computable and even in NP thanks to general ideas from game theory. However all algorithms proposed are exponential. I will review the fundamental results on these games: existence of a unique pure and positional optimal strategy, its caracterisation, the fixed point formulation ... I will also sketch some of the best algorithms, which are all very simple to describe but often hard to analyze.

2013-06-20Johanna Franklin (University of Connecticut, USA)Lowness in recursive model theory
We say that a Turing degree is low in a particular context if using it as an oracle in that context has the same result as using no oracle at all. Lowness has been studied in the frameworks of degree theory, learning theory, and randomness. I will discuss lowness in recursive model theory: we say that a degree is low for isomorphism if, whenever it can compute an isomorphism between two recursively presented structures, there is actually a recursive isomorphism between them. I will describe the class of Turing degrees that are low for isomorphism, identify some particular subclasses, and show how it behaves with respect to measure and category.

This work is joint with Reed Solomon.
2013-06-27Thomas Johnstone (New York City College of Technology, USA)Indestructibility Results for Ramsey Cardinals
We will prove basic indestructibility results for Ramsey cardinals. Ramsey cardinals can be characterized by the existence of certain nontrivial elementary embeddings–Ramsey embeddings–whose domains are certain transitive sets of size \(\kappa\). However, the standard lifting techniques do not quite apply: Ramsey embeddings have domains and targets that need not be closed (not even under countable sequences), and so the usual diagonalization method to build generic filters does not apply. Moreover, to verify that the lifted embedding witnesses that \(\kappa\) is Ramsey, one has to show (among other things) that the ultrafilter generated by the lifted embedding is countably complete. We will present a new diagonalization criterion for models without closure, and we will also present sufficient conditions so that the ultrafilter generated by the lift is countably complete. As a result, we will show that Ramsey cardinals are indestructible by a variety of forcing notions.

This is joint work with Victoria Gitman.
2013-06-28Santi Spadaro (Silesian University of Opava, Czech Republic)Chain conditions for topological bases
We will speak about joint work with Menachem Kojman and Dave Milovich on a class of topological cardinal invariants capturing base properties studied by the Russian school since the seventies. We will discuss analogies and differences between these cardinal invariants and the "cellularity" of a topological space, that is the supremum of the sizes of families of pairwise disjoint non-empty open sets. We will discuss their behavior in products and independence phenomena regarding their value on box products.
2013-10-10Stéphane Le Roux (TU Darmstadt, Germany)Determinacy, a two-way bridge from logic to economics
Borel determinacy gives a game-theoretic meaning to Borel sets and is invoked, e.g., to describe the Wadge hierarchy; finite-memory determinacy of Muller games and positional determinacy of parity games both connect logic and automata theory. All these various determinacy results involve two-player games with two possible outcomes saying who wins. First, I will show that all such determinacy results can be generalised for many outcomes instead of two; second, I will show that Borel determinacy can also be generalised for many players and many outcomes instead of two; third, I will mention a possible further generalisation of Borel determinacy that is not proved yet; and finally I will explain how a new result on finite tree-games might be generalised on the Baire space.

The three abstracts below give slightly more details:



2013-10-17Victor Selivanov (A.P. Ershov Institute of Informatics Systems, Novosibirsk, Russia and Siberian Branch of the Russian Academy of Sciences)Some Hierarchies of $\mathsf{QCB_0}$-Spaces

In this joint work with Mathias Schroeder we define and study hierarchies of topological spaces induced by the classical Borel and Luzin hierarchies of sets. Our hierarchies are divided into two classes: hierarchies of countably based spaces induced by their embeddings into the domain \(P_\omega\), and hierarchies of spaces (not necessarily countably based) induced by their admissible representations (in the sense of computable analysis). We concentrate on the non-collapse property of the hierarchies, on the relationships between hierarchies in the two classes, and on the relationship with the Kleene-Kreisel continuous functionals.
2013-10-24Masahiko Sato (Kyoto University, Japan)Mathematics as open-ended human activities
In this talk, I will explain how I view mathematics as open-ended human activities. I have gradually developed this view through my experience of designing and implementing a proof assistant which I call NF (Natural Framework).

NF is based on a theory of classes, which is rigorously describable in an informal metalanguage but is never completely formalizable since it is an open-ended theory. I will sketch the outline of the theory and the philosophy behind it.

NF has the following characteristics.

(1) NF is implemented on top of a simple programming language called Ez.

(2) Ez is implemented on top of its minimal sublanguage.

(3) Ez supports definitions and constructions of new finitary mathematical objects from scratch.

(4) Ez objects are either mutable or rigid (i.e., immutable).

(5) The background theory of NF/Ez is finitary, but in this finitary framework one can design and define any conceivable formal systems.

(6) Because of (3) above, NF/Ez is open-ended and extendable by the users.

I will also compare NF with other proof assistants and will argue that open-endedness of the system is essential to assist developing mathematics as open-ended human activities.
2013-10-31Tin Lok Wong (KGRC)Some model theory of Peano arithmetic
First, I would like to share with the audience some of my favourite theorems in my area of research. They should give an idea of why Peano arithmetic (PA) is a good theory. Second, I will explain what it means for a cut (i.e., an initial segment) of a model of PA to be ‘generic’. I will show why (I think) genericity is a good notion for cuts.
2013-11-07Barnabas Farkas (KGRC)Some problems related to Borel ideals: Towers, Luzin-type families, forcing (in)destructibility, and more
I will discuss the possible existence of (maximal) towers in Borel ideals. I will prove the following results: ''Theorem 1.'' After adding uncountable many Cohen reals, there are towers in every tall Borel P-ideals. ''Theorem 2 (Brendle).'' It is consistent that there are no towers in any Borel P-ideals.

Related to Theorem 2, I will show that although we cannot expect more, namely "domination" from \(\sigma\)-centered forcing notions, the Localization forcing (which is \(\sigma\)-\(n\)-linked for every \(n\)) dominates every Borel P-ideals. Here the reverse implication is still an open problem. 

Theorem 2 will lead us to the next natural question, namely the possible existence of so-called idealized Luzin-type families of size \(\omega_2\). To obtain such a family (very probably) we need some kind of iterated destruction of the ideal without Cohen reals (at least). I will show that the Random forcing works for the summable ideal but not for the density zero ideal.
2013-11-14Pavel Semukhin (KGRC)Automatic models of first-order theories
An automatic structure is an algebraic structure whose operations and relations can be recognized by finite automata. In the first part of this talk I'll provide a general introduction to automatic structures. The second part will be devoted to our joint work with Frank Stephan on model theoretical properties of automatic structures. We will consider the questions like (1) If a complete theory ''T'' has only countably many countable model (in particular, if ''T'' is uncountably categorical) and one of the models of ''T'' is automatic, are all the other models of ''T'' automatic? (2) Fix ''N''>=1; is there a complete theory that has exactly ''N'' automatic models? (3) If the countable saturated model of ''T'' is automatic, does this imply that the prime model of ''T'' is also automatic?
2013-11-21Giorgio Venturi (Scuola Normale Superiore di Pisa, Italy)Generalized side conditions
In this talk I would like present the method of generalized side conditions, first proposed by Neeman in 2011: a method that allows to give uniform consistency proofs for the existence of objects of size \(\aleph_2\). Generally speaking a poset that uses models as side conditions is a notion of forcing whose elements are pairs, consisting of a working part which is some partial information about the object we wish to add and a finite \(\in\)-chains of elementary substructures of \(H(\theta)\) (for some regular cardinal \(\theta\)) whose main function is to preserve cardinals. I will present in details the pure generalized side conditions poset and I will briefly present the poset that allows to force a club in \(\omega_2\), the poset for forcing a Thin Tall Boolean algebra and the one for forcing an \(\omega_2\) Souslin tree. In the end I will present a generalization of the combinatorial principle P-Ideal Dichotomty (PID) to ideals of uncountable sets, that I called PID\(_{\aleph_1}\), sketching the consistency proof of one instance of PID\(_{\aleph_1}\). If I will have time I will also discuss the possibility to generalize this method and its link with the problem of generalizing the Forcing Axioms.

=== References ===

[1] Itay Neeman: "Forcing with sequences of models of two types". Preprint.

[2] Boban Veličković and Giorgio Venturi: "Proper forcing remastered". In ''Appalachian Set Theory'' (Cummings, Schimmerling, eds.), LMS lecture notes series, 406, 331–361, 2012.
2013-11-28Asylkhan Khisamiev (Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences)Universal functions over locally finite structures
Now it is generally accepted that one of the important generalizations of the concept of computability is \(\Sigma\)-definability (generalized computability) in admissible sets. This generalization has made possible to study computability problems over arbitrary structures, for instance, over the field of real numbers. A crucial result of classical computability theory is the existence of an universal partially computable function. It is known that an universal \(\Sigma\)-predicate exists in every admissible set, but this is false for \(\Sigma\)-functions. Therefore, it is interesting to know which conditions on \(\mathfrak{M}\) guarantee the existence of an universal \(\Sigma\)-function in the hereditarily finite admissible set \(\mathbb{HF}(\mathfrak{M})\) over \(\mathfrak{M}\). In this talk we discuss the problem of the existence of an universal \(\Sigma\)-function in the hereditarily finite admissible set over some locally finite structures.
2013-12-05Andrea Medini (KGRC)Seven characterizations of non-meager P-filters
We will begin with an introduction to topological notions of homogeneity. For example, a space is countable dense homogeneous if for every pair \((D,E)\) of countable dense subsets of \(X\) there exists a homeomorphism \(h:X\longrightarrow X\) such that \(h[D]=E\). Then, we will gradually move to the study of the topology of filters on \(\omega\), focusing on ultrafilters and non-meager filters. Here, we identify a filter with a subspace of \(2^\omega\) through characteristic functions. The following is joint work with Kenneth Kunen and Lyubomyr Zdomskyy.

Recall that a filter is a P-filter if it contains a pseudointersection of each one of its countable subsets. An ultrafilter that is a P-filter is called a P-point. While Shelah showed that it is consistent that there are no P-points, it is a long standing open problem whether it is possible to construct a non-meager P-filter in ZFC. We will give several topological/combinatorial conditions that, for a filter on \(\omega\), are equivalent to being a non-meager P-filter. In particular, we show that a filter is countable dense homogeneous if and only if it is a non-meager P-filter. This answers a question of Hernández Gutiérrez and Hrušák. Along the way, we also strengthen a result of Miller.

Finally, we will show that the statement "Every non-meager filter contains a non-meager P-subfilter" is independent of ZFC (more precisely, it is a consequence of \(\mathfrak{u}<\mathfrak{g}\) and its negation is a consequence of \(\Diamond\)). It follows from results of Hrušák and Van Mill that, under \(\mathfrak{u}<\mathfrak{g}\), the only possibilities for the number of types of countable dense sets of a non-meager filter are \(1\) and \(\mathfrak{c}\).

2013-12-12Oleg Gutik (Ivan Franko National University of Lviv, Ukraine)Topological semigroups: embeddings and Čech-Stone compactifications
The problem of an embedding of some (topological) semigroups into compact-like topological semigroups will be discussed. Also, we shall give the sufficient conditions on a (semi-)topological semigroup \(S\) under which the Čech-Stone compactification \(\beta S\) preserves algebraic and topological properties of \(S\).
2013-12-19Michael Hrušák (Universidad Nacional Autónoma de México)Strong measure zero in metric spaces and Polish groups
We shall study the notion of strong measure zero in a general setting of a separable metric space. We shall investigate to what extent can the theorem of Galvin, Mycielski and Solovay be extended to an arbitrary Polish group. We shall also see that there are two classes of separable metric spaces on which the notion of strong measure zero behaves differently. The dividing line being provided by the so called small ball property.

This is joint work with W. Wohofsky and O. Zindulka.