The Andrzej MostowskiCentenary Conference

The Program Committee

Krzysztof R. AptJacek CichonJohann MakowskyVictor W. Marek (chair)Damian NiwinskiAndrzej PelcMarcin SabokMarian SrebrnyJouko VäänänenJan Wolenski

The Organizing Committee

Krystyna JaworskaMirosław KurkowskiHenryk MichalewskiFilip MurlakDamian NiwinskiMarcin SabokMarian SrebrnySzymon Torunczyk

PUBLISHED BY THE ORGANIZING COMMITTEE OF THE ANDRZEJ MOSTOWSKI CENTENARY

CONFERENCE

MOSTOWSKI100.MIMUW.EDU.PL

PHOTOGRAPHS COURTESY OF TADEUSZ MOSTOWSKI

11-13 October 2013

Contents

1 Conference schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1 All lectures listed in tables 4

2 Friday . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1 Lectures 6

2.2 Soirée musicale 8

3 Saturday . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.1 Lectures 10

3.2 Rump session 12

4 Sunday . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.1 Lectures 13

5 How to get around . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5.1 By public transportation 15

5.2 By taxi 15

5.3 On foot 15

5.4 On bicycle 15

All lectures listed in tables

1 — Conference schedule

1.1 All lectures listed in tables

Time Friday8:45-9:00 Opening9:00-10:00 W. Hugh Woodin

The Mostowski Collapse andthe Inner Model Program

10:00-10:20 coffee break10:20-11:20 Thomas Jech

Measure algebras11:30-12:30 Menachem Magidor

Inner Models constructedfrom generalized logics

12:30-14:30 lunch14:30-15:30 Angus Macintyre

The Elementary Theory ofthe Adele Ring over a Number Field

15:30-15:50 coffee break15:50-16:50 Mikołaj Bojanczyk

Computation in Sets With Atoms17:00-18:00 Harvey Friedman

Concrete mathematical incompleteness18:00-20:00 welcome party

20:00- Soirée musicale

1.1 All lectures listed in tables 5

Time Saturday9:00-10:00 Christian Rosendal

Large scale geometry of metrisable groups10:00-10:20 coffee break10:20-11:20 Mirna Džamonja

Would Mostowski have liked it?11:30-12:30 Sławomir Solecki

General approach to finite Ramsey theoryand a new Ramsey theorem

12:30-14:30 lunch14:30-15:30 Zlil Sela

Low dimensional topologyand the elementary theory of groups

15:30-15:50 coffee break15:50-16:50 Jan Wolenski

Tarski, Mostowski and philosophy(not only of mathematics)

17:00-18:20 Rump session19:30-21:00 reception21:00-22:00 Johann A. Makowsky

Andrzej Mostowski, the man and his legacy

Time Sunday9:00-10:00 Ludomir Newelski

Topological methods in model theory10:00-10:20 coffee break10:20-11:20 Krzysztof Krupinski

Polish structures11:30-12:30 Anand Pillay

Transcendence, differential equations,and model theory

12:30-14:30 lunch14:30-15:30 Yiannis Moschovakis

Effective Descriptive Set Theory15:30-15:50 coffee break15:50-16:50 Leszek Kołodziejczyk

The importance of approximate countingin bounded arithmetic

17:00-18:00 Jouko VäänänenThe many lives of generalized quantifiers

LecturesSoirée musicale

2 — Friday

2.1 Lectures8:45 - 9:00 Opening9:00 - 10:00 W. Hugh Woodin, University of California, Berkeley.

Title: The Mostowski Collapse and the Inner Model Program.Abstract: The Mostowski collapse is ubiquitous in modern Set Theory. One importantmanifestation is in condensation principles which lie at the core of the Inner Model Pro-gram. These condensation principles are really generalizations of the Mostowski collapseand the basic open problems of the Inner Model Program can be reformulated in terms ofsuch principles.

10:00 - 10:20 coffee break

10:20 - 11:20 Thomas Jech, Pennsylvania State University and Czech Academy of Sciences.Title: Measure algebras.Abstract: A measure algebra is a complete Boolean algebra carrying a countably additivemeasure. We give a description of measure algebras in Boolean algebraic terms. The workwas inspired by a problem of John von Neumann that he stated in 1937 in the Scottish book.

11:20 - 11:30 break

11:30 - 12:30 Menachem Magidor, Hebrew University.Title: Inner Models constructed from generalized logics.Abstract: This talk combines two subjects which were very much in the centre ofMostowski’s interests: Generalized logics and quantifiers and Set Theory. In a jointwork with J. Kennedy and J. Väänänen we study the inner models of Set Theory that areobtained like the constructible universe where each stage of the construction is obtainedfrom the previous stage by taking all definable subsets of the last stage. We get a richcollection of inner models by changing the notion of “definable” from “first order definable”to “definable by some generalized logic”. The study of these inner models yields someinteresting inner models and it leads to some intriguing problems.

2.1 Lectures 7

12:30 - 14:30 lunch in the building of the Department

14:30 - 15:30 Angus Macintyre, Queen Mary College, University of London.Title: The Elementary Theory of the Adele Ring over a Number Field (joint work withJamshid Derakhshan).Abstract: We greatly refine work of Weisspfenning on the adeles by internalizing to ringtheory the Feferman–work Vaught on restricted products (inspired in turn by Mostowski’swork on products). In particular we get very precise quantifier–eliminations, using ad-vanced work on the model theory of Henselian fields. We prove measurability of definablesets, but show that definable is not the same as locally closed. We present some extensionsboth of the Feferman-Vaught work, and of the Weisspfenning work, inspired by quadraticreciprocity and more general reciprocity theorems. We pay attention to uniformity over allnumber fields.

15:30 - 15:50 coffee break

15:50 - 16:50 Mikołaj Bojanczyk, University of Warsaw.Title: Computation in Sets With Atoms.Abstract: Sets with atoms (also known as permutation models, or Fraenkel-Mostowskisets) were introduced in set theory by Fraenkel in 1922, and further developed byMostowski. In the last decade, they were rediscovered for the computer science commu-nity, where they are called nominal sets. Nominal sets are an active research topic in thesemantics of programming languages.This talk is about a research programme, which studies a notion of finiteness which onlymakes sense in sets with atoms, called “orbit-finiteness”. The research programme isto see what happens to discrete mathematics when sets are replaced by sets with atoms,and finiteness is replaces by orbit-finiteness. Two examples of this research programme,motivated by computer science, are orbit-finite versions of programming languages, andof Turing machines.Programming languages. In sets with atoms, the set of rational numbers with their orderis an orbit-finite set. Therefore, in a programming language for sets with atoms, one canwrite the following program, which exhaustively searches through all triples of rationalnumbers, searching for a counter-example to transitivity of the order:

counterexample:=false

for x in Q do

for y in Q do

for z in Q do

if (x < y) and (y < z) and not (x < z) then

counterexample:=true

This programme will be executed in finite time, and will never set counterexample totrue.Turing machines. When Turing machines with orbit-finite state spaces and alphabets areconsidered, the computability landscape becomes richer, in the sense that there are fewerequivalences between models. For instance, P is provably different from NP (although thisresult is unlikely to shed any light on the real P vs NP problem). Also, nondeterministicdecidability does not imply deterministic decidability (unlike for normal Turing machines,where the two notions coincide).

2.2 Soirée musicale 8

16:50 - 17:00 break

17:00 - 18:00 Harvey Friedman, Ohio State University.Title: Concrete mathematical incompleteness.Abstract: Mathematicians view mathematics as a special subject with singular attractivefeatures. Most feel that the great power and stability of the “rule book for mathematics”is an important component of their relationship with mathematics. They believe that therules are so powerful and stable that they do not have to remember or refer to it.

The Incompleteness they see does not shake their confidence in this stability, as the sub-ject matter or nature of the underlying objects, or both, are of very great distance - in afundamental sense – from that of the mathematics they are familiar with.

I initiated Concrete Mathematical Incompleteness (CMI) in the late 1960s with the ideathat the future of Incompleteness depends on its success. Here we are 45 years later. Idiscuss the continually evolving examples of CMI with many mathematicians, includingtop luminaries such as A. Connes, C. Fefferman, H. Furstenburg, T. Gowers, M. Gromov,D. Kazhdan, Y. Manin, B. Mazur, D. Mumford. All of these luminaries are fully aware ofwhat is at stake, and had interesting reactions.

Above all, it is obvious that they do not identify, in any way, “fundamental or importantmathematics” with “mathematics people have done or are doing no”. My examples werejudged strictly on fundamental mathematical standards of simplicity, naturalness, intrinsicinterest, and depth. This is fortunate since the integration of CMI with existing concretemathematical developments will likely occur only at later stages of CMI.

The outer limits of CMI live in the Borel measurable sets and functions between Polishspaces. New examples live in the finite sets of positive integers under addition only, andare explicitly Π0

1. The corresponding systems range from 2 quantifier induction to thehuge cardinal hierarchy. Some historical highlights are: long finite sequences, continuouscomparability of countable sets of reals, Kruskal’s and Higman’s theorem, Graph MinorTheorem, Borel Diagonalization, Borel determinacy, Borel selection, and Boolean RelationTheory. The main focus of the talk will be on the current state of CMI.

18:00 - 20:00 welcome party

2.2 Soirée musicaleHarvey Friedman, piano recital. The event will take place in the Department building, room2180. The event starts at 20:00.

Johann Sebastian BachJesu, Joy of Man’s Desiring, from cantata Herz und Mund und Tat und Leben,transcription by Myra Hess

Wolfgang Amadeus MozartSonata no. 11 in A major, KV 331Andante graziosoMenuettoAlla Turca

2.2 Soirée musicale 9

Ludwig van BeethovenSonata no. 8 in C minor Pathétique, Op. 13Grave – Alegro di molto e con brioAdagio cantabileRondo: Allegro

Franz SchubertImpromptu no. 3 in G-flat major, Op. 90

Intermission

Fryderyk ChopinNocturne no. 2 in E-flat major, Op. 9Nocturne no. 20 in C-sharp minor, Op. posth.

Grande valse brillante in E-flat major, Op. 18Valse in D-flat major Minute, Op. 64 no. 1Valse in C-sharp minor, Op. 64 no. 2

Fantaisie-Impromptu in C-sharp minor, Op. posth.

Johannes BrahmsRhapsody no. 2 in G minor, Op. 79

John Philip SousaThe Stars and Stripes Forever, transcription by Vladimir Horovitz

LecturesRump session

3 — Saturday

3.1 Lectures9:00 - 10:00 Christian Rosendal, University of Illinois at Chicago.

Title: Large scale geometry of metrisable groups.Abstract: Large scale geometry of finitely generated or locally compact groups has longbeen one of the cornerstones of geometric group theory and its connections with harmonicand functional analysis. However, many of the groups of interest in logic, topology andanalysis fail to be locally compact, such as automorphism groups of countable struc-tures, diffeomorphism and isometry groups. For these there has been no canonical wayof defining their large scale structure, as it is possible, e.g., with the word metric on afinitely generated group. Moreover, recently many groups have turned out to have nonon-trivial large scale structure at all, despite being non-compact. We present a theory oflarge scale structure of metrisable groups and among other things determine the necessaryand sufficient conditions for this structure to be unique up to coarse or quasi-isometricequivalences. Applications to model theory will be presented.

10:00 - 10:20 coffee break

10:20 - 11:20 Mirna Džamonja, University of East Anglia, Norwich.Title: Would Mostowski have liked it?Abstract: : We shall speak of some points of interest in set theory and applications overthe last years and overlaps of this with model theory and foundations.

11:20 - 11:30 break

11:30 - 12:30 Sławomir Solecki, University of Illinois at Urbana-Champaign.Title: General approach to finite Ramsey theory and a new Ramsey theorem.Abstract: In recent years, finite Ramsey theorems have been used in studying dynamicsof groups of interest to Model Theorists and Descriptive Set Theorists. I will present ageneral approach to finite Ramsey theory, which reveals the formal algebraic structureunderlying results of that theory. Most concrete unstructured finite Ramsey results turn outto be special instances of the main theorem of this general theory. I will also describe a

3.1 Lectures 11

very recent concrete Ramsey result—dual Ramsey theorem for trees—which was obtainedusing the general theory. This result was partly inspired by the internal logic of the theoryand partly by considerations related to amenability of groups.

12:30 - 14:30 lunch in the building of the Department

14:30 - 15:30 Zlil Sela, Hebrew University, Jerusalem.Title: Low dimensional topology and the elementary theory of groups.Abstract: We will survey some results on the first order theory of free and free productsof groups, and indicate concepts and techniques from low dimensional topology that playan essential role in proving, and at times even stating, some of these results. These willinclude Tarski’s problem on the elementary equivalence of non-abelian free groups, andVaught-Malcev problem on the elementary equivalence of free products of pairs of elemen-tarily equivalent groups, as well as other results on the structure of definable sets in relatedtheories. We will assume no prior knowledge in model theory nor in low dimensionaltopology.

15:30 - 15:50 coffee break

15:50 - 16:50 Jan Wolenski, Jagiellonian University.Title: Tarski, Mostowski and philosophy (not only of mathematics).Abstract: The Warsaw logical milieu in the interwar period was compact as far as theissue concerns the basic direction of research and views about logic and its tasks, but itconsisted of philosophers (Lesniewski and Łukasiewicz as the main figures) and math-ematicians (Tarski, Lindenbaum and Mostowski as leading persons). This division isfuzzy to some extent, because all Warsaw logicians shared interests in philosophy andwere ready to discuss philosophical problems not necessarily belonging to philosophyof mathematics. Yet almost all members of this group (Lesniewski was an exception inthis respect) considered logic as independent of philosophy. In particular, formal logicalinvestigations and results should not be limited by philosophical presuppositions or preju-dices. On the other hand, philosophical inspiration for logic was welcomed. Both Tarskiand Mostowski represented this attitude in a particularly paradigmatic manner. Sincetheir philosophical remarks were short and not numerous, it is possible to reconstructtheir more general philosophical views. For example, both were inclined to empiricismas an epistemological position and favored conceptualism and even nominalism (Tarski)as an ontological view about the existence of mathematical objects. The paper is devotedto looking for other philosophical ideas possibly present in works of Tarski and Mostowski.

16:50 - 17:00 break

17:00 - 18:20 The rump session – see the program in the next section

19:30 - 21:00 reception in the Kazimierzowski Palace, the main campus on Krakowskie Przed-miescie 26/28

21:00 - 22:00 Johann A. Makowsky, Technion, Haifa (the lecture will take place in the Kazimier-zowski Palace).Title: Andrzej Mostowski, the man and his legacy.Abstract: I will try to remember A. Mostowski, the human being, the scholar, the bridge

3.2 Rump session 12

builder, and his legacy, both in the context of the Cold War and from a personal perspective.

3.2 Rump session

17:00-17:10 Angel Garrido, UNEDOn Fuzzy Logic.

17:10-17:20 Piedad Yuste, UNEDIndian Logic.

17:20-17:30 Ali Enayat, University of GothenburgLesser known gems from Mostowski.

17:30-17:40 Erdogan Sen, Istanbul Technical UniversityAsymptotic Formulas for Eigenvalues and Eigenfunctions of a Discontinuous Sturm–Liouville Type Problem with Retarded Argument.

17:40-17:50 Victor Marek, University of KentuckyMostowski and Computer Science.

17:50-18:00 Jan Dobrowolski, Uniwersytet WrocławskiSmall compact G-rings.

18:00-18:10 Andrzej Salwicki, Cardinal Stefan Wyszynski University in WarsawHalting property of a version of Euclid’s algorithm is independent from Peano’s axioms ofnatural numbers.

18:10-18:20 Kolya Umanets, Saint Petersburg State UniversityModel-theoretic analogue of Berkovich space.

This is a tentative program. The final program is on the website

MOSTOWSKI100.MIMUW.EDU.PL/DOKU.PHP?ID=RUMP

Lectures

4 — Sunday

4.1 Lectures9:00 - 10:00 Ludomir Newelski, Uniwersytet Wrocławski

Title: Topological methods in model theory.Abstract: I will survey the development of topological methods in model theory. I willdiscuss Morley rank and forking in stable theories and also various ways to generalize it tothe unstable theories. I will focus on topological dynamics. Topological dynamics yieldsgeneral counterparts of the notion of a generic type in a stable group.

10:00 - 10:20 coffee break

10:20 - 11:20 Krzysztof Krupinski, Uniwersytet Wrocławski.Title: Polish structures.Abstract: I will define the notion of Polish structure introduced by myself a few yearsago, and discuss counterparts of some fundamental notions from stability theory in thecontext of Polish structures. Then I will focus on structural results about groups and ringsin the context of Polish structures, some of which were obtained in collaboration with F.Wagner or J. Dobrowolski.

11:20 - 11:30 break

11:30 - 12:30 Anand Pillay, University of Leeds.Title: Transcendence, differential equations, and model theory.Abstract: I give an application of model theory, specifically the theory DCF0 of differen-tially closed fields, to the problem of understanding algebraic relations between solutionsof ordinary differential equations belonging to the Painleve family. I will give the requiredbackground as well as describing elementary aspects of the proof (joint with my student J.Nagloo).

12:30 - 14:30 lunch in the building of the Department

14:30 - 15:30 Yiannis Moschovakis, UCLA.

4.1 Lectures 14

Title: Effective Descriptive Set Theory.Abstract: Andrzej Mostowski first noticed the similarities between the classical Boreland projective sets of real numbers and Kleene’s arithmetical and analytical sets of naturalnumbers and used them to define the hyperarithmetical sets. These “analogies” betweenthe two subjects, as they were first called, eventually developed into a common “effective”theory of Borel, analytic and projective sets on Polish spaces, a common extension of theclassical and the recursive hierarchies which illuminates and enriches them both.

15:30 - 15:50 coffee break

15:50 - 16:50 Leszek Kołodziejczyk, University of Warsaw.Title: The importance of approximate counting in bounded arithmeticAbstract: Bounded arithmetic is a collective name for a family of relatively weak theoriesof first-order arithmetic in which the exponential function is not total and the inductionscheme can only be applied to bounded formulas. Such theories, which were given theirmodern form by Sam Buss in the 1980’s, are studied largely because of their connections tocomputational complexity and propositional proof complexity. However, they are also ofinterest from a more traditional foundational point of view, as a way of understanding howmuch of finite mathematics can be developed without exponentiation or other “infeasiblyfast-growing” functions.The fundamental problem in bounded arithmetic is whether the induction scheme forthe class of all bounded formulas can be finitely axiomatized. This problem and relatedquestions appear to be beyond the reach of current methods, so much effort has goneinto the study of relativized theories, for which unprovability and separation results aremore accessible. I will discuss the two main open problems on the frontier of research onrelativized bounded arithmetic, and explain recent work suggesting that a single theoryplays a major role in the context of both problems. The theory in question, axiomatizedby a fragment of the induction scheme and a version of the so-called weak pigeonholeprinciple, was introduced by Emil Jerabek in order to formalize the notion of approximatecardinality of definable sets.The talk will be based on joint work with Sam Buss, Neil Thapen and Konrad Zdanowski.

16:50 - 17:00 break

17:00 - 18:00 Jouko Väänänen, University of Helsinki and University of Amsterdam.Title: The many lives of generalized quantifiers.Abstract: Generalized quantifiers have thrived in logic, linguistics and computer sciencein ways perhaps unanticipated by Mostowski who introduced them in 1957. They havecome to manifest a perfect symbiosis between model theory and set theory. Their modeltheoretic properties reflect a rich array of set-theoretical concepts such as large cardinalsand combinatorial principles. I will end with a few remarks on dependence logic as a newway of thinking about generalized quantifiers.

By public transportationBy taxiOn footOn bicycle

5 — How to get around

5.1 By public transportationYou may decide for a one-ride ticket, a 20,40 or 60 minutes ticket, a one-day ticket or a three-days ticket. You are expected to stamp your ticket immediately after boarding. Some busesand trams are equipped with a vending machine. The best source of information about theWarsaw transportation system is the website JAKDOJADE.PL with an accompanying free andpaid apps for the main phone platforms. The website will indicate not only the route, but also therecommended type of ticket.

5.2 By taxiThe day tariff is roughly 2 złoty per kilometer and is always indicated on the windows of everyWarsaw cab. This amounts to a bill below 40 złoty for a typical ride inside of Warsaw. You mayorder a taxi by phone dialing a number 00 48 221919.

5.3 On footSeveral walking maps are presented on the conference website.

5.4 On bicycleThere is a city-wide system of bike rentals available to everyone registered at VETURILO.WAW.PL.The rental is free until 20 minutes and costs 1 złoty until 1 hour. In practice most destinations indowntown Warsaw are reachable in less than an hour.

Sponsors

Clay Mathematics InstituteUnited States National Science FoundationEuropean Mathematical SocietyPresident of the Polish Academy of SciencesThe Committee on Philosophical Sciences of the Polish Academy of SciencesCopernicus Center KrakówWarsaw Center of Mathematics and Computer ScienceInternational Union of History and Philosophy of Science –

Division of Logic, Methodology and Philosophy of ScienceIOS PressAssociation for Symbolic Logic

Organizing Institutions

Department of Mathematics, Informatics, and Mechanics, University of WarsawPolish Mathematical SocietyInstitute of Mathematics, Polish Academy of Sciences