Papers are usually presented in .dvi, .ps and .pdf form. Click on the title for the .dvi file, on .ps for the PostScript file, and on .pdf for the Acrobat file.

Should users find that any of the files here are corrupt, please inform the author. At present --- March 2009 --- email addresses for him are

ardm@univ-reunion.fr
ardm@email.mathematik.uni-freiburg.de
ardm@dpmms.cam.ac.uk

ARTICLES ON AXIOMATICS:

Readers might find this brief commentary on the paper helpful.

[Note that both the .dvi file and the PostScript file are arranged for double-sided printing and will generate successively an unnumbered title page containing the abstract, AMS classification numbers and keywords, and the author's current address, then a blank unnumbered page, then on page ii a table of contents of the paper and on page iii a chart giving the axioms of the various systems discussed; and finally the main text on pages numbered 4--88.]

Slim models of Zermelo Set Theory Journal of Symbolic Logic 66 (2001) 487--496. (A companion paper, exploring the weakness of Zermelo's original system for recursive constructions. 7 pp of A4 plain TeX.) .ps .pdf

A Term of Length 4,523,659,424,929 Synthese 133 (2002) 75--86 (A calculation of the number of symbols required to give Bourbaki's definition of the number 1; to which must be added 1,179,618,517,981 disambiguatory links. The implications for Bourbaki's philosophical claims and the mental health of their readers are discussed.) .ps .pdf

Weak systems of Gandy, Jensen and Devlin in Set Theory: Centre de Recerca Matemàtica, Barcelona 2003-4 edited by Joan Bagaria and Stevo Todorčević, Trends in Mathematics, Birkhäuser Verlag, Basel, 2006, 149-224. .ps .pdf

(contains a variety of constructions proving the independence of various natural statements in various weak systems of set theory and shedding light on flaws in Devlin's treatise Constructibility. The first third of the paper develops versions of the model building techniques of my paper Slim Models, applicable to the systems considered in the paper. The heavily syntactic middle third of the paper examines the effect of adding the axiom ``the class of finite subsets of any set is a set" to the various systems proposed by Devlin and by Gandy; and closes with the suggestion that the addition, instead, of the slightly weaker form ``for each positive integer k the class of subsets of size k of any given set is a set", mightly be precisely the elusive optimal strengthing of Devlin's system BS for the purposes to hand. The final third returns to model-building mode to answer other questions about these systems.)

A note on the schemes of replacement and collection Archive for Mathematical Logic, 46 (2007) 43-50. .ps .pdf

(derives all the axioms of ZF from a scheme that is apparently weaker than either the collection or the replacement scheme.)

Epireflections and supercompact cardinals (to appear in the Journal of Pure and Applied Algebra ) .ps .pdf (with Joan Bagaria and Carles Casacuberta)

(Casacuberta and others have used Vopěnka's principle to prove that certain functors admit representations by local functions. We weaken the set-theoretical hypothesis required, in the case that the functors satisfy stricter conditions than before. 15 pages, available from the arXiv preprint data base at arXiv:math.CT/0703119.)

Unordered pairs in the set theory of Bourbaki 1949 .ps .pdf

(We construct a supertransitive model of Bourbaki's 1949 system for set theory, which is a subsystem of Zermelo set theory less the pairing axiom but with axioms for ordered pairs and for cartesian products. In our model, ordered pairs are available, and the corresponding cartesian product of two sets is a set, but there are failures of the principles that the unordered pair of two sets is a set and that the union of two sets is a set.)

Work in progress on RUDIMENTARY RECURSION, PROVIDENT SETS AND FORCING:

Rudimentary recursion and provident sets .ps .pdf

(We introduce the class of rudimentarily recursive functions, and study those sets, which we call provident, which are transitive and closed under all rudimentarily recursive functions, allowing parameters from within the set in question. We identify a single rudimentary recursion, with parameter, to instances of which all others reduce; we obtain various characterizations of provident sets, showing in particular that the segment $J_\nu$ of the Jensen hierarchy is provident if and only if $\omega\nu$ is an indecomposable ordinal; and we find strong uniform bounds on the rate of growth of rudimentarily recursive functions. )

Provident sets and rudimentary set forcing .ps .pdf

(Using the theory of rudimentary recursion and provident sets developed in the previous paper, we give a treatment of set forcing appropriate for working over models of a theory PROVI which may plausibly claim to be the weakest set theory supporting a smooth theory of set forcing, and of which the minimal model is Jensen's $J_\omega$. Much of the development is rudimentary or at worst given by rudimentary recursions with parameter the notion of forcing under consideration. Our development eschews the power set axiom. We show that the forcing relation for restricted wffs is propagated through our hierarchies by a rudimentary function, and we show that the construction of names for the values of rudimentary and rudimentarily recursive functions is similarly propagated. Our main result is that a set-generic extension of a provident set is provident. )

Set forcing over models of Zermelo or Mac Lane (to appear in the Proceedings of the Zermelo Centenary meeting "One Hundred Years of Axiomatic Set Theory" held at Brussels in October 2008.) .ps .pdf

(We discuss the problem of forcing over a transitive model of Zermelo set theory, or alternatively of Mac Lane set theory. We identify difficulties when the model fails to be provident. We show that the provident closure of a model of M or Z is itself a model of M or Z. We show that for provident models of these theories, the operations of forming generic extensions and lunes commute.)

ARTICLES ON DYNAMICS:

Delays, recurrence and ordinals Proceedings of the London Mathematical Society, (3) 82 (2001) 257--298. (using set-theoretical ideas to study the iteration of derived omega-limit sets in dynamical systems, proves that, from every starting point, that iteration stabilises not later than the first uncountable ordinal, gives examples in Baire and in Cantor space for each countable ordinal of iterations lasting exactly that long, gives an example of a recursively defined point starting from which the iteration stabilises at the first non-recursive ordinal, and gives new examples of complete analytic sets. ) .ps.pdf

Recurrent points and hyperarithmetic sets (gives details and further results omitted from "Delays". In: Set Theory, Techniques and Applications, Curacao 1995 and Barcelona 1996 conferences, edited by C. A. Di Prisco, Jean A. Larson, Joan Bagaria and A. R. D. Mathias, Kluwer Academic Publishers, Dordrecht, Boston, London, 1998, 157--174.) .ps.pdf

Analytic sets under attack (answers questions left open in "Delays" by constructing two recursive points, a and b , in Baire space such that the second derived omega-limit set starting from a is a complete analytic set whilst the third is empty, whereas starting from the point b the iteration of derived omega-limit sets stabilises exactly at the first uncountable ordinal, yielding yet another complete analytic set. To appear in the Mathematical Proceedings of the Cambridge Philosophical Society.) .ps.pdf

Choosing an attacker by a local derivation Acta Universitatis Carolinae - Math. et Phys., 45 (2004) 59--65. .ps .pdf

A scenario for tranferring high scores Acta Universitatis Carolinae - Math. et Phys., 45 (2004) 67--73. .ps .pdf

ARTICLE ON NUMBER THEORY:

On a conjecture of Erdos and Cudakov (establishes a simple case of an unsolved problem. In: Combinatorics, Geometry and Probability: Proceedings of the conference dedicated to Paul Erdos on the occasion of his 80th birthday, edited by Bollobas Bela et al, Cambridge University Press (1993), 487--488.) .ps.pdf

ESSAYS IN THE SOCIOLOGY OF MATHEMATICS:

The Ignorance of Bourbaki (a commentary on the foundational stance of the Bourbaki group. In: Mathematical Intelligencer 14 (1992) 4--13 MR 94a:03004b, and also in Physis Riv. Internaz. Storia Sci (N.S.) 28 (1991) 887--904, MR 94a:03004a. A translation by Andras Racz into Hungarian is available, under the title Bourbaki tevutjai, in A Termeszet Vilaga, 1998, III. kulonszama.) .ps.pdf

La Ignorancia de Bourbaki (A translation by José Maria Almira Picazo into Spanish, published in La Gaceta de la Real Matemàtica Espanola, 7, (2004), no. 3.) .ps .pdf

Further remarks on Bourbaki (A reply to criticism by Professor Sanford L. Segal of the above essay.) .ps.pdf

What is Mac Lane missing ? (a comment on the foundational stance of Saunders Mac Lane; MR 94g:03010; published with a reply by Mac Lane, MR 94g:03011, in Set Theory of the Continuum, ed. H. Judah, W.Just, H.Woodin; Mathematical Sciences Research Institute Publications Volume 26, Springer-Verlag, 1992.) .ps.pdf

Logic and Terror (an account of the position of formal logic in Russia during the first forty years of the Soviet regime); in Jahrb. Kurt-Goedel-Ges. (1990) 117--130 MR 92m:01039, and also , in a longer version, in Physis Riv. Internaz. Storia Sci (N.S.) 28 (1991) 557--578 MR 93d:03014) .ps.pdf

watch this space.

EXPOSITORY ARTICLE:

Strong Statements of Analysis (Discusses examples of natural statements concerning irrational numbers that are equivalent, provably in ZFC, to strong set-theoretical hypotheses. 12 pp of A4 plain TeX. Published by the Bulletin of the London Mathematical Society 32 (2000) 531--526, with a comment by Mac Lane on page 527, but in a version that contains over 100 alterations made by the Editor of that organ in defiance of the author's wishes. The present piece is the authentic text.) .ps.pdf

Brief Remarks on the Axiom of Choice (three pages of non-technical comment on the status and limitations of the Axiom of Choice in mathematics: intended as cultural background for Part II mathematicians) .ps.pdf

FINALLY:

A list of the author's other mathematical publications is available, and this whole web-site in French. The author is the Directeur Scientifique of the research group ERMIT at the Université de la Réunion.

The Équipe ERMIT in Réunion
• The Bonn-Vienna Guide to Mathematical Logic around the World
• The Logic Links of Tübingen
• The Florida Directory of home pages of set theorists, maintained by Jean Larson
• The Budapest List of email addresses of set theorists, maintained by Péter Komjáth
• The Omaha Directory of home pages of set theorists, maintained by Andrzej Roslanowski

  The Foundations of Mathematics discussion group, moderated by Stephen Simpson in Pennsylvania

• The Erdös Number home page at Oakland
• The Shelah Archive at Rutgers

  The Paris and Davis entrances to the preprint collection at Los Alamos

• ZentralBlatt
• Sommaires Bordelais of contents of mathematical journals

Reports before truncation: ermfull1 ermfull2 ermfull3 ermfull4

Reports after truncation: ermcut2 ermcut3 ermcut4

Other reports: irerep1 irerep2 irerep3 irerep4 irerep5 irerep6