Maurice Boffa's 60th birthday Workshop
Program

Abstracts

• Georges HANSOUL (Liège): Décoration booléenne de graphes
  Un espace est dit pseudo-indécomposable s'il n'est pas somme de deux espaces, tous deux non homéomorphes à lui.Il est dit primitif s'il est compact, métrique et admet une base d'ouverts fermés pseudo-indécomposables.Des exemples sont donnés par l'espace triadique de Cantor, les ordinaux dénombrables non limites et les algèbres de Lindenbaum de nombreuses théories usuelles. Les espaces primitifs peuvent être étudiés par le biais de décorations, id. d'une fonction d définie sur un graphe et à valeur dans les ouverts-fermés de l'espace de façon à ce que d(x) soit déterminé canoniquement par les d(y), où (x,y) est une arête du graphe. Comme pour les ensembles, on peut donner une construction de décorations de graphes non bien-fondés par approximations successives.

• Véronique BRUYERE (Mons): Automates et systèmes de numération
  Le théorème de Cobham dit qu'un ensemble d'entiers reconnaissable par automate fini dans deux bases multiplicativement indépendantes est nécessairement une union de progressions arithmétiques. Depuis 1985, année où j'ai étudié ce théorème dans mon mémoire de fin d'études, beaucoup de travaux ont été publiés (généralisations à des bases non entières, approches par la logique, par les substitutions ...) dont plusieurs sont issus de l'équipe de logique de l'UMH. Je compte faire le point sur ce thème dans mon exposé.

• Roland HINNION (Bruxelles): Tree-properties for ordered sets
  The notion of “tree” (and “well-pruned tree”) of height kappa, with theta-finite levels (for kappa, theta cardinals) can be naturally generalized to become the one of “theta-tree on a partial order (E,<=)”; so the classical “tree-property” (or “ramifiability”) inspires several variants of “tree-properties for orders”. The most interesting cases concern directed orders D, have applications to compactness problems [4] and fixed point problems [5] for set-continuous operators, and present interesting links with “large cardinals” [7]; one can show that they cannot be reduced simply to (classical) tree-properties for the main parameters of D, namely the “characteristic cardinal” (the largest delta such that any delta-finite subset of D has an upper bound) and the “cofinality” (the least possible cardinal of a cofinal subset of D). Several “combinatorial criteria” have been established [7,8], some of which use not necessarily directed orders, motivating investigation of “tree-properties for (partial) orders”; reasonable sufficient conditions [8], necessary conditions [9] and in some cases (as: theta=aleph₀; E finite; E countable) more “geometric” characterizations appeared [9]. Several very natural questions however are still not completely solved; for example: “ramifiability” for a cartesian product of directed sets, with respect to adequate “ramifiability” of the factors; or “absolute well-pruned ramifiability” (i.e. relative to all well-pruned trees on E) for countable orders.

  [4] Hinnion, R. Ramifiable directed sets. Math. Logic Quarterly 44, 216-228 (1998).
  [5] Dzierzgowski, D.; Esser, O. and Hinnion, R. Fixed-points of set-continuous operators. Math. Logic Quarterly 46 (2000)
  [7] Esser, O. and Hinnion, R. Large cardinals and ramifiability for directed sets. Math. Logic Quarterly 46 (2000)
  [8] Esser, O. and Hinnion, R. Combinatorial criteria for ramifiable ordered sets (submitted)
  [9] Esser, O. and Hinnion, R. Tree-properties for ordered sets (submitted)

• Françoise POINT (Mons): Groupes satisfaisant une identite et propriete de Milnor
  In the first part of the talk we will try to give an overview of some results on groups satisfying an identity. Then, we will give a generalization of the Milnor property (introduced by J.Milnor to show that a finitely generated soluble group with that property is polycyclic) and prove a nilpotency criteria in the class of finitely generated soluble group (extending a result of G. Endimoini).

• Olivier ESSER (Bruxelles): The consistency strength of a positive set-theory
  The aim of this talk is to present the theory GPK+infty. This theory has a comprehension scheme for bounded positive formulas and an axiom scheme of closure which behaves like a topological closure. Our main result about this theory is that it is mutually interpretable with KM+ “On has the tree-property”; KM is the Kelley-Morse class-theory; “On has the tree-property” is the natural translation to the class of ordinals of the corresponding notion for cardinals in ZF. Another interesting result on GPK+infty is that the axiom of choice is inconsistent with it.

• Alexis BES (Paris): La conjecture d'Erdos-Woods
  Julia Robinson demandait en 1949 si l'on peut définir au premier ordre l'addition et la multiplication dans la structure <IN;S,cop>, où S désigne la fonction successeur et cop(x,y) est interprété par “x et y sont premiers entre eux”. Cette question est toujours ouverte. Je parlerai des réponses partielles proposées (indépendamment) par Woods et Richard dans les années 80, et en particulier du résultat de Woods qui établit un lien étroit entre la question de J.Robinson et une conjecture difficile de théorie des nombres, dite conjecture d'Erdös-Woods.

• Arnaud MAES (Mons): Entrelacs Brunniens et généralisations d'après Stanford
  A family of brunnian braids can easily be constructed using a commutator collection process. T. Stanford recently proved that a variation of this process characterize brunnian braids (and generalizations), and gives an algorithm for deciding whether a given braid is brunnian. We present Stanford's results.

• André PETRY: Faire de l'Analyse non standard sans Logique ?
  Comment peut-on faire de l'Analyse non standard sans connaissance spéciale en Logique, en se basant sur une méthode introduite par Keisler.

• Dirk VAN DALEN (Utrecht): Foundations of Brouwer's Intuitionism
  Already at the time of his dissertation, Brouwer had a more or less coherent philosophical basis not only for his mathematics, but for 'everything'. That is to say, science in general, language, social behaviour, etc. His views on the mathematical universe was determined by the reflections on human mind and consciousness. In particular the non-lawlike nature of the mathematical objects was dictated by his so-called "causal sequences". The properties of the continuum and other objects of mathematical practice were based on an analysis of the underlying sequence structure. A number of consequences will be demonstrated, such as the extremely connected nature of the continuum, and even of the irrationals and similar sets. From the logical point of view, the intuitionistic structures of real life ask for a highly refined analysis. This can be seen already in simple theories such as that of equality.

• Serge GRIGORIEFF (Paris): Automates et mots transfinis
  Les langages de mots transfinis reconnus par automate ont été introduits par Büchi pour prouver la décidabilité de la théorie monadique des ordinaux <= aleph1. Nous étudions les relations de mots transfinis. En particulier, nous montrons que l'uniformisation des relations rationnelles est possible pour les langages de mots de longueur <omega^n mais échoue à partir de l'ordinal omega^omega.

• Ulrich FELGNER (Tübingen): Fonctions régressives
  In einer Vorbemerkung gehe ich auf die Geschichte des beruehmten Satzes von Neumer und Fodor ein. Im ersten Hauptteil gebe ich sodann eine Anwendung des Satzes von Neumer-Fodor auf den Automorphismenturm einer unendlichen, zentrumsfreien Gruppe. Im zweiten Hauptteil gebe ich unter Verwendung regressiver Funktionen eine neue Konstruktion eines Aronszajn-Baumes.

• Henk BARENDREGT (Nijmegen): Set theory versus type theory as foundation
  A foundation for mathematics is an axiomatic system such that most of mathematics can be formalised in it. This formalisation can be done either in principle or actually. There are reasons why an 'in principle' foundation is useful. Set theory provides such a foundation. There are reasons why an 'actual' foundation is useful. Type theory provides such a foundation.

• Marco FORTI (Pisa): Wanted: a strongly comprehensive theory of collections
  We present an axiomatization of the primitive notions of collection and set, stemming from the foundational programme of E.De Giorgi. This axiomatization is intended to capture the most general concept underlying the naive notions of class or aggregate, as conceived by Frege and Cantor. In our view, these notions try to mediate the somewhat different concepts of “extension of an arbitrary property” and of “content of a finite list”. The former is captured in our theory by the notion of collection, and the latter by that of set =“small manageable collection”. Cantor's set theory, later axiomatized by Zermelo, isolates sets as “not too big” collections, which can be freely manipulated. Von Neumann's axiomatization of sets and classes identifies “sethood” and “elementhood”, a Limitation of Size Principle being the basic criterion for both. We consider all collections, big and small, as first class objects, and not merely as a “façon de parler”. Hence we introduce the collection Coll of all collections and consider various “Gödel Operations” acting on Coll: pairing, difference, cartesian product, etc. The idea that sets are “small” is embodied in the Axiom of Replacement, while their “simple and controlled internal structure” is axiomatized by assuming that the graph of membership between sets and objects is a collection.
  Although strong Comprehension Principles can be derived, in the usual Bernays' style, we cannot apply consistently to arbitrary collections all the manipulations carried out in ordinary mathematical practice. In particular union, intersection, and cartesian product of a collection of collections may not exist as collections (and so does the collection of all subcollections). These constructions play an important rôle in almost all areas of Mathematics, Logic and Semantics, and it seems appropriate to introduce operations Un, Int, Cart, Sub_coll that carry out the intended tasks, so as to make substantial use of collections. So far we have only proved the consistency of the weaker axiom:
  Un,Int act on every set of collections and every collection of sets. Cart acts on every set of collections.
  We conjecture (at least) that arbitrary unions and intersections are indeed consistent, and we pose the question to the attention of all interested scholars.

• Elisabeth BOUSCAREN (Paris): Théorie des modèles et Conjecture de Manin-Mumford (d'après Ehud Hrushovski)
  Nous présenterons des applications récentes de la Théorie des Modèles à des questions de Géométrie Diophantienne sur les corps de nombres. Nous indiquerons en particulier comment E.Hrushovski, en utilisant la Théorie des corps algébriquement clos munis d'un automorphisme, donne une nouvelle démonstration de la conjecture de Manin-Mumford, démonstration qui produit de bonnes bornes effectives.

Hotel

The conference hotel is the LIDO Hotel.

How to reach the Conference places ?

• Thursday: ULB, Campus de la Plaine, local 2NO906
  From the airport the best to reach ULB is to go to Brussels Central by train and from there to take the Metro as follows: You have to take the metro in the direction HERMANN DEBROUX and to stop at the station DELTA (it is direct, around 15 minutes). From the upperlevel of this sation you have to follow the signs ULB (this implies that you will take on the right an underground passage (around 200 meters long), you will exit in a small square with trees. You are on the Campus de la Plaine of the ULB (close to VUB, the Flemish University at Brussels) In front of you , you will have a way (straightahead) going through a square (named Forum - the university restaurant lies on the right side of the square). From the square, follows the signs NO.
  The seminar room is on the ninth floor of the building NO
  

• Friday: UMH, local 0A11 (Pentagone)
  
• Saturday: UMH, Grands Amphithéâtres

Registration Form TO SEND BACK TO christian.michaux@umh.ac.be
First Name & Name:
I will attend the meeting on

Thursday 23 14h30-18h30	(Brussels)	Yes	No
Friday 24 10h-13h	(Mons)	Yes	No
Saturday 25 10h-18h30	(Mons)	Yes	No

I will take part to the following:

Dinner on March 23	Yes	No	(place and price not yet fixed)
Lunch on March 24	Yes	No	(price around 400 BEF)
Lunch on March 25	Yes	No	(free)
Conference Dinner on March 25	Yes	No	(price around 1100 BEF)

I will attend the Honoris Causa ceremony:

Yes

No

If you need hotel reservation in Mons, please send me a mail (price per night ranges from 1800 BEF to 3500 BEF).
Warning: it is quite urgent to reserve for the meals and the ceremony.

Back to Mathematical Logic at the University of Mons-Hainaut
This page has been accessed times.