Abstracts

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Tutorials

Boban Velickovic: Transfinite Games.

Andres Villaveces: The Model Theory of Sheaves.

This is a self-contained tutorial with emphasis on the Fundamental Theorem (a generalization of both the Forcing Theorem and the Łoś Theorem) and on constructions of examples - classical tori, Hilbert Spaces over their spectra, etc.

This minicourse looks at connections between model theory and topology and geometry and is suitable for a wider mathematical audience. Depending on interest, there may be a short foray into philosophical aspects of the model theory of sheaves as well, perhaps during the final session.

Talks

Peter Aczel: Rudimentary Constructive Set Theory

Constructive Set Theory (CST) provides an extensional purely set-theoretical, but intuitionistic logic based and constructively acceptable, setting for the formalisation of the practice of constructive mathematics. An axiom system for CST was first introduced in a 1975 JSL paper by John Myhill. A key feature of his axiom system was the use of the Exponentiation axiom for function spaces instead of the Powerset axiom. A related axiom system, CZF, has the same theorems as ZF when the logic is made classical, but has the low proof theoretic strength of the extension ID_1 of Peano Arithmetic.

Rudimentary CST (RCST) is a very weak fragment of CZF that is, nevertheless, strong enough to express many standard non-numerical set-theoretical constructions that do not involve powersets or even function spaces.

I will describe RCST and its metatheory and, if there is time, some arithmetical extensions.

Joan Bagaria: Structural Reflection

We will consider some forms of reflection in the universe of all sets akin to Vopenka's Principle and we will show they are equivalent to the existence of some well-known large cardinals, such as supercompact or extendible cardinals.

John Baldwin: Complexity and Absoluteness in $L_{\omega_1,\omega}$

Translation from a sentence of $L_{\omega_1,\omega}$ to an associated atomic class is a key tool for the study of categoricity in infinitary logic. We compare the complexity of definition of model theoretic notions such as $\omega$ -stability and excellence for a sentence of $L_{\omega_1,\omega}$ and the associated atomic class. We show these properties are $\Sigma^1_2$ on the sentences and $\Pi^1_2$ on the classes. (Lower bounds have not been established in the atomic class case). In either case these properties are absolute. But the question of whether $\aleph_1$ -categoricity is absolute remains open for either formulation. Connecting this study with more classical descriptive set theory we show that the class of models of a sentence of $L_{\omega_1,\omega}$ whose automorphism groups admit a complete left invariant metric is $\Pi^1_2$ but not $\Sigma^1_2$ . Techniques from model theory, recursion theory and descriptive set theory are used. Much of this is joint work with David Marker.

Zoé Chatzidakis: Some recent results on finite approximate subgroups.

I will present and discuss some recent results on near (or approximate) subgroups by Hrushovski, and by Breuillard, Green and Tao. One says that a finite subset X of G is a k-near subgroup of G if the cardinality of $XX^{-1}X$ is bounded by k times the cardinality of X. This is the case for instance if X is a subgroup of G.

Mirna Dzamonja: SUSIFA, a new kind of forcing axiom

Forcing axioms relating to $\omega_1$ have formed a vibrant area of research in set theory. Some of these axioms can be generalised to other successors of regular cardinals but $\omega_1$ . However, none of them applies to successors of singular cardinals. Not even a simplest forcing notion, such as the one for adding a Cohen real can be generalised directly to the successor of a singular cardinal. This is of course due to the known limitations in failing the Singular Cardinal Hypothesis and a need of the involvement of large cardinals. In joint work with Saharon Shelah that this talk is based upon, we introduce a forcing axiom which applies to successors of singular cardinals. We apply it to the problem of the existence of universal graphs at such cardinals and some other problems.

Pietro Galliani: Dependencies in terms of constancy and epistemic operators

I decompose the dependency atoms into constancy atoms and announcement operators, and show that both of these ingredients are necessary for filling the step from First Order Logic to the Dependence Logic developed by Jouko Väänänen.

Lauri Hella: A Game for Characterizing the Size of Formulas with Generalized Quantifiers

The complexity of first-order properties of models is often measured in terms of the quantifier rank of formulas needed for expressing them. This is mostly because the standard Ehrenfeucht-Fraž¨ssé game gives a useful tool for determining this measure. However, quantifier rank is an extremely rough measure of complexity: the number of non-equivalent formulas of quantifier rank q is an exponential tower of height q with a polynomial (depending on the vocabulary) on top.

In a joint paper with Jouko Väänänen we consider the size of formula as an alternative measure of complexity for first-order properties of models. We introduce a game that characterizes the equivalence of models with respect to all formulas of a given size w. Using a natural restrction of this game, we are able to prove optimal bounds on the size of existential formulas for defining some simple first-order properties. In the talk I will consider an extension of the game that works for the extension of first-order logic with monotone generalized quantifiers.

Åsa Hirvonen: Generalized isomorphisms in metric model theory

In metric model theory the genuine isometric isomorphisms are not always the only natural ones to study. When studying Banach spaces, functional analysts are often more interested in linear isomorphisms, which are allowed to change the norm. In the talk I will give furhter examples illustrating the emergence and benefits of generalized notions of isomorphism. I will also present a way to treat them model theoretically. The talk is based on joint work with Tapani Hyttinen.

Wilfrid Hodges: Historical remarks on dependence and scope.

Jouko Väänänen in his book 'Dependence Logic' gives novel and sophisticated logical tools for describing dependences. How did logicians handle dependence in the days before modern tools were available? Most were unaware of the notion, or else avoided it. I will examine some discussions in the 11th century logician Ibn Sina, who persistently pointed out the importance of two-quantifier sentences and raised some problems about them connected with scope and Skolem functions. It would be good if we could show that Jouko's work would have helped Ibn Sina solve his problems, but I can't promise this. Ibn Sina's logic never reached the West; but if time allows, I will mention how this work of Ibn Sina probably influenced the use of logical scope in the legal rulings of Ayatollah Khomeini.

Tapani Hyttinen: Generalized Baire Space

For an uncountable regular $\kappa$ , letting the sets $U_\eta=\{\xi\in \kappa^{\kappa}\mid \eta\subset\xi\},\,\eta\in\kappa^{<\kappa}$ , be a basis for a topology on $\kappa^{\kappa}$ , one gets a natural topological space, a generalized Baire pace (intersection of $\kappa$ -many dense open sets is dense). From the very beginning of 90's, Jouko Väänänen has both studied the space and kept the topic alive through the times when it has not been popular. In my survey I will start from the beginning of 70's and R. Vaught and end with some of the resent developments and ongoing research.

Meeri Kesälä: Interpreting groups and fields in non-elementary classes

Geometric stability theory is one of the most vibrant branches of model theory and a source of many interesting applications. Finitary abstract elementary classes were developed in the authors PhD thesis in Helsinki to provide a suitable framework for generalizing geometric stability theory beyond first order definable classes. We generalize a result of Hrushovski about interpreting groups and fields on the geometry induced on the realizations of a regular type. This is joint work with Tapani Hyttinen.

Peter Koepke: Checking Natural Language Proofs

According to the standard foundational view, mathematical statements correspond to first-order formulas, and mathematical proofs correspond to formal derivations. So first-order logic naturally provides a semantics for the common language of mathematics. The linguistic study of mathematical texts aims at identifying mechanisms by which natural, intuitive and argumentative phrases can be interpreted formally. In the talk we present a reverse engineering approach: the Naproche system (Natural language proof checking, www.naproche.net) uses methods of natural language processing (NLP) and automatic theorem proving (ATP) to analyse mathematical texts written in a controlled and restricted natural language and and to check the logical correctness of proofs. We give examples from Edmund Landau's "Foundations of Analysis" and Euclid's "Elements". The Naproche system can be seen as a natural language interface to formal mathematics, with a range of possible applications.

Phokion G. Kolaitis: Random Graphs and the Parity Quantifier

The classical zero-one law for first-order logic asserts that the asymptotic probability of every first-order definable property of finite graphs always exists and is either zero or one. Over the years, zero-one laws have been established for numerous extensions of first-order logic, including least-fixed point logic, finite-variable infinitary logics, and certain fragments of existential second-order logic. The zero-one law, however, fails to hold for any logical formalism that is powerful enough to express parity, that is, the property "there is an odd number of elements"; in fact, for such logical formalisms, even the convergence law fails to hold. In this work, we turn the parity barrier into a feature and systematically investigate the asymptotic probabilities of properties of finite graphs expressible in first-order logic augmented with the parity quantifier. Our main result is a "modular convergence law" that captures the limiting behavior of properties expressible in this extension of first-order logic on finite graphs. This is joint work with Swastik Kopparty (MIT).

Juha Kontinen: Definability in Dependence Logic

Dependence Logic is a new logic that incorporates the concept of dependence into first-order logic. Dependence Logic was introduced by Jouko Väänänen in his monograph "Dependence Logic" (2007). The expressive power of dependence logic coincides with that of existential second-order logic. Since formulas of dependence logic express properties of sets of assignments, not properties of individual assignments, this result does not directly extend to open formulas. We show that open formulas of dependence logic correspond to the negative fragment (downwards monotone) of existential second-order logic. We also show (generalizing a result of John P. Burgess) that the negation of dependence logic is not a semantic operation, that is, knowing the class of models that satisfy a formula does not completely determine the class of models of its negation. This is joint work with Jouko Väänänen.

Roman Kossak: Why do we study nonstandard models of arithmetic?

I will give several answers to the question on the title and I will provide some justifications.

Michał Krynicki: On logical properties of arithmetics in finite domain

I will consider theories of sentences true in almost all initial segments of standard models for various arithmetics. I will investigate decidability and axiomatizability as well as other properties of these theories.

Oskari Kuusela: Wittgenstein’s philosophy of logic

In this paper I discuss Wittgenstein’s conception of the status of logical calculi and of other related logical/philosophical tools (such as exact definitions) that can be employed to analyse and clarify language use. Wittgenstein’s conception of logic is compared with Carnap’s Hilbertian conception with the purpose of highlighting distinctive features of the former and explaining how Wittgenstein’s conception can be used to solve problems that arise for Carnap.

Menachem Magidor

Jaroslav Nesetril

Ilkka Niiniluoto: Constructivist Realism in Mathematics

Traditional classifications of the main schools in the philosophy of mathematics are based upon two questionable presuppositions. First, it is assumed that a realist, who wishes to defend objective truth values of mathematical statements, has to be either a Platonist or a physicalist. Secondly, a constructivist, who regards mathematical entities as human constructs rather than pre-existing objects, has to be either a subjective mentalist or an objective idealist. In contrast to these alternatives and their many variants, this paper argues that it is possible to be a constructivist and a realist at the same. Mathematical entities are public creations in the human-made domain of abstract artefacts that Karl Popper called World 3. At the same time, they are not completely transparent to their makers, so that mathematical constructs are full of open problems for us to study. The applicability of mathematics to physical, mental, and social reality can be explained by the Theory of Measurement, which also helps to show what is wrong with the empiricist approaches to mathematical knowledge.

Jeff Paris: Pure Inductive Logic

The aim of my talk will be to demarcate a subfield of Carnapian Inductive Logic, which I'm calling Pure Inductive Logic, and explain with a couple of examples how I see it differing from Carnap's vision of an Applied Inductive Logic. In short Pure Inductive Logic retains Carnap et al's basic formulation but investigates its mathematics consequences in the absence of any intended interpretations. Because of this it avoids the many, some say fatal, criticisms that in Philosophy have been leveled against Carnap's ideas of logical probability and Inductive Logic and becomes in spirit akin to Set Theory.

Krister Segerberg: Trying to model metaphor

A simple kind of metaphor can be given a (rudimentary) treatment within the modal logic of belief change.

Stevo Todorcevic

Scott Weinstein: Preference based on reasons

We describe a logic of preference in which modal connectives reflect reasons to desire that a sentence be true. Various conditions on models are introduced and analyzed.

This is joint work with Daniel Osherson (Princeton).

Dag Westerstahl: Consequence Mining: A New Approach to Logical Constants

We explore the idea of getting at the logical constants, or at least the constants, of a language by 'extracting' them from a given consequence relation, taken as primitive. Under minimal assumptions, we show that (a) this gives the expected constants for familiar consequence relations in logic, and (b) the extraction operation is an inverse, in a suitable sense, to the Bolzano-Tarski style method of defining a consequence relation from any set of symbols taken as constants. This is joint work with Denis Bonnay.

Boris Zilber: Between model theory and physics

Modern physics is full of mathematically dubuous methods of calculations that magically produce correct testable results. Model theory has means to address these issues. I am going to discuss some of the related questions.

Last modified: Mon Sep 13 12:02:26 EEST 2010