Depository of papers by A. R. D. Mathias

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 2013 --- email addresses for him are

Recent additions are marked by

On a generalisation of Ramsey's theorem Introduction 1. 2. 3. 4. 5. 6. 7. 8. 9. (Introduction: what are now called non-Ramsey sets were then called Scott families. At the time I wrote the thesis, I knew of Prikry's result that all Borel sets are Ramsey, but not that Galvin had also proved it. ¶¶ 1, 2, and 7 are largely devoted to expounding, in the language of Boolean-valued models, Solovay's construction of his celebrated model, his own account not at the time being available. ¶¶ 3, 4, 5, 6, 8 and 9 are original to me, except where stated. The key concept of a condition capturing a dense set is introduced in ¶ 4, on page 66, and applied in pages 72-77 to prove the ``all subsets generic" property for "plain Mathias" forcing, and later, on page 80, to establish the Prikry property for this forcing. ¶¶ 5 and 6 digress from the main proof; ¶ 5 explores properties of Mathias reals, and has recently borne fruit by showing, answering a question of Zapletal, that if R is an analytic equivalence relation on the power set of omega in which every equivalence class is countable, then, ``almost everywhere", xRy implies that the symmetric difference of x and y is finite. ¶ 6 compares Mathias reals with other kinds of generic real known at the time. ¶ 7 analyses collapsing algebras; in ¶ 8 it is shown that in Lévy's model obtained by collapsing an inaccessible all definable families are Ramsey, and then in ¶ 9 MacAloon's method is followed to obtain "all sets Ramsey" in an appropriate submodel.

Notes on set theory Madison notes

The Strength of Mac Lane Set Theory Annals of Pure and Applied Logic, 110 (2001) 107--234. .ps.pdf

[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) .pdf

Set forcing over models of Zermelo or Mac Lane, in One Hundred Years of Axiomatic Set Theory, Cahiers du Centre de Logique, 17, ed. Roland Hiunnion and Thierry Libert, Academia Bruylant, Louvain-la-Neuve (Belgium) (2010) pages 41-46. .ps .pdf

A Term of Length 4,523,659,424,929 Synthese 133 (2002) 75--86 .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

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

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

Epireflections and supercompact cardinals Journal of Pure and Applied Algebra 213 (2009) 1208-1215. .ps .pdf (with Joan Bagaria and Carles Casacuberta) A sequel, "Definable orthogonality classes in accessible categories are small", with J. Bagaria, Carles Casacuberta and Jiří Rosický) is about to appear in J. European Math. Soc. (It extends the main results of "Epireflections and supercompact cardinals" from absolute epireflective subcategories to definable reflective subcategories, by showing that the existence of a proper class of supercompact cardinals implies that each object in a Sigma_2-definable class of objects of an accessible category has a small subobject in the same class. That in turn shows that each Sigma_2-definable class of objects is bounded, and thence that each absolute orthogonality class is a small-orthogonality class. Hence, each reflection in an accessible category whose class of local objects is absolute is an F-localisation for some set F of morphisms.)

Rudimentary recursion and provident sets .ps .pdf (with Nathan Bowler) (submitted)

Provident sets and rudimentary set forcing .ps .pdf (accepted; revision in progress)

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

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

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 tévútjai, in A Természet Világa, 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 ?

Here is a scan of both articles: mathiasmaclane.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

Hilbert, Bourbaki and the scorning of logic .ps.pdf

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

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


Reports before truncation: ermfull1 ermfull2 ermfull3 ermfull4

Reports after truncation: ermcut2 ermcut3 ermcut4

Other reports: irerep1 irerep2 irerep3 irerep4 irerep5 irerep6