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One Hundred Years of Axiomatic Set Theory

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111 pages

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This Cahier du centre de logique is made of papers presented at the homonymous conference that was organized by the editors in October 2008 to celebrate the 100th birthday of axiomatic set theory. It can virtually be divided into two bundles of papers : one discussing systems related to Zermelo's, the other dealing more specifically with systems related to type theory and stratification, both of which treating of key notions in the axiomatization of set theory.

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Ajouté le 01 juin 2012
Nombre de lectures 51
EAN13 9782296493056
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ONE HUNDRED YEARS OF
AXIOMATIC SET THEORYDej a parus
Cahier 1 epuise
Intuitionnisme et theorie de la demonstration.
Cahier 2 epuise
Textes de Jean Pieters.
Cahier 3 epuise
J.-L. Moens, Forcing et semantique de Kripke{Joyal.
Cahier 4 epuise
La theorie des ensembles de Quine.
Cahier 5 epuise
T. E. Forster, Quine’s New Foundations.
Cahier 6 epuise
Logique et informatique.
Cahier 7 epuise
L’antifondation en logique et en theorie des ensembles.
Cahier 8
Ph. de Groote (ed.), The Curry{Howard Isomorphism.
Cahier 9
A. Petry (ed.), Methodes et analyse non standard.
Cahier 10 epuise
M. R. Holmes, Elementary Set Theory with a Universal Set.
Cahier 11
Chr. Michaux (ed.), De nability in Arithmetics and Computability.
Cahier 12
P. Van Praag, Aspects de la dualite en mathematique.
Cahier 13
O. Esser, Une theorie positive des ensembles.
Cahier 14
K. De Clercq, Logica in communicatie.
Cahier 15
S. Richard, La conception semantique de la verite.
Cahier 16
M. Crabbe and T. E. Forster (eds), Proceedings of the 70th anniversary
NF meeting in Cambridge.
Une version electronique des volumes epuises sera prochainement
disponible sur http://www.logic-center.be/cahiers.CAHIERS DU CENTRE DE LOGIQUE
17
ONE HUNDRED YEARS
OF AXIOMATIC SET THEORY
Roland HINNION and Thierry LIBERT (eds)
Centre National de Recherches de Logique
Nationaal Centrum voor Navorsingen in de Logica
LOUVAIN-LA-NEUVE 2010
Log17FT 1 2/16/10, 11:55 PMCAHIERS DU CENTRE DE LOGIQUE
Directeur de la collection :
M. Crabbe
Comite de redaction :
D. Batens, M. Crabbe, J. De Greef, Ph. de Groote (Nancy),
D. Dzierzgowski, T. E. Forster (Cambridge), R. Hinnion,
M. R. Holmes (Boise), Th. Lucas, J. Meheus,
Chr. Michaux, A. Petry.
Cahier 17 edite par :
Roland Hinnion et Thierry Libert,
Universite libre de Bruxelles.
Secretariat :
D. Servais
Composition :
D. Dzierzgowski
Centre national de recherches de logique
http://www.logic-center.be/cahiers
D/2010/4910/9 ISBN 978{2{87209-974{0
c BRUYLANT-ACADEMIA s.a.
Grand-Place 29
B{1348 Louvain-la-Neuve
Tous droits de reproduction, d’adaptation ou de traduction, par quelque procede que
ce soit, reserves pour tous pays sans l’autorisation de l’editeur.
Imprime en Belgique.Foreword
In response to the paradoxes of naive set theory, axiomatic foundations for
set theory and mathematics were proposed in 1908 in the following two
papers :
I Zermelo, Untersuchungen ub er die Grundlagen der Mengen Lehre, Ma-
thematische Annalen, 30, 261{281.
I Russell, Mathematical logic as based on the theory of types, American
Journal ofs, 30, 222{262.
This seventeenth volume of the Cahiers du Centre de logique is devoted
to set-theoretic systems related to Zermelo’s, such as fragments of ZF,
but also to Simple Type Theory or even to Quine’s ‘New Foundations’
| after all, Quine was born in 1908 too ! It is essentially made of papers
presented at the homonymous conference that was organized by the editors
in Brussels (ULB) on 30{31 October 2008.
These have been arranged so that this Cahier can virtually be divided
into two bundles of papers : one discussing systems related to Zermelo’s
(Halbeisen, Pettigrew, Mathias, Hinnion), the other dealing more speci-
cally with systems related to type theory and strati cation (Hinnion,
Kaye, Forster). That barrier is permeable and researchers in those elds
would be the rst to admit that any strict division is futile.
More important is the fact that all the papers that compose this volume
will nally treat of key notions in the axiomatization of set theory, such
as choice (Halbeisen), in nity (Pettigrew), foundation (Hinnion), typing
(Forster), as well of famous model constructions, such as forcing (Mathias)
and models with automorphisms (Kaye).We wish of course to thank the authors for their contributions, and Daniel
Dzierzgowski for the typesetting. The realization of this volume would
not have been possible either without the support of the Belgian Centre
national de recherches de logique.
The editors,
Roland Hinnion & Thierry Libert.
One Hundred Years of Axiomatic Set Theory, 1908–2008
30–31 October 2008, Brussels (ULB)
I Thursday October 30th
– 10h00{11h00 : Roland Hinnion, Some speci cities of Zermelo’s system
– 11h30{12h30 : Adrian Mathias, Forcing over models of Zermelo set
theory
– 14h30{15h30 : Lorenz Halbeisen, On models of Zermelo’s system in
which Zermelo’s axiom fails
– 16h00{17h00 : Philip Welch, The Inner Model Hypothesis
I Friday October 31th
– 10h00{11h00 : Thomas Forster, Paris{Harrington in NF
– 11h30{12h30 : Richard Pettigrew, In nity and separation in the foun-
dations of arithmetic and analysis
– 14h30{15h30 : Richard Kaye, Automorphisms and constructions of mo-
dels of set theoryContents
L. Halbeisen
Comparing cardinalities in Zermelo’s system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
R. Pettigrew
The foundations of arithmetic in nite bounded
Zermelo set theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21
A. R. D. Mathias
Set forcing over models of Zermelo or Mac Lane. . . . . . . . . . . . . . . . . . . . . . . . .41
R. Hinnion
Some speci cities of Zermelo’s Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
R. W. Kaye
Automorphisms and constructions of models of set theory . . . . . . . . . . . . . . . 73
R. W. Kaye
On the bounding lemma for KF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .89
T. E. Forster
The Paris-Harrington Theorem in an NF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .97Cahiers du Centre de logique
Volume 17
Comparingcardinalities in
Zermelo's system
by
L. Halbeisen
Universität Zürich
Abstract. The aim of this note is to define and compare cardinalities in
Zermelo’ssystemof1908(withoutusingtheAxiomofChoice). Themaintoolto
comparecardinalities inSetTheory(withorwithouttheAxiomofChoice)isthe
Cantor-Bernstein Theorem. We shall present two different proofs of the Cantor-
Bernstein Theorem and shall give some historical background. In particular,
we shall see that this theorem was proved independently by Cantor, Dedekind,
Bernstein, Korselt, Zermelo, and Peano, but not by Schröder —even though it
is sometimes cited as Schröder-Bernstein Theorem.
1. Axiomatisations of Set Theory
In order to define the notion of cardinality in a formal way, we first need
some axioms. So, let us start by presenting a few axiomatic systems of
Set Theory.
In1908,Zermelopublishedin[33]hisfirstaxiomaticsystemofSetTheory,
which we shall denote by ZC, consisting of the following seven axioms:
(a)Axiom der Bestimmtheit
which corresponds to the Axiom of Extensionality,