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
(my Peterhouse Fellowship (1969) and Cambridge Ph.D. (1970) dissertation, written in July 1968 in Bonn-Beuel; the "paragraphs" (really, the chapters) are scanned separately as .pdf files.)
(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
(a scan of the notes of my 1969 lectures on admissibility and constructibility at the University of Madison, Wisconsin)
The Strength of Mac Lane Set Theory
Annals of Pure and Applied Logic, 110
(A study of set-theoretic systems related to
topos theory and classical set theory. Version of 15 March, 2001. iii + 85 pp of A4 plain TeX.
A careful summary of its contents is given by Bell in his review
MR 2002g:03105, and readers might also find this commentary on the paper helpful.
A recurrent theme of the paper is that the axioms of Kripke-Platek set theory may, without increasing the consistency strength, be added to those of Mac Lane or Zermelo by passing to what in my my Brussels paper is called the lune of the ground model; for many purposes the more explicit process, introduced there and studied in my paper with Bowler, of passing instead to its provident closure might prove sufficient. The paper shows that Z + AC is indeed consistent relative to Zermelo's system Z, but the inadequacy, demonstrated in Slim Models, of Z for recursive constructions necessitates an oblique approach. The paper applies techniques of Kaye and Forster to yield the first published proof of the theorem implicit in Kemeny's thesis, that the simple theory of types, with infinity, is equiconsistent with Mac Lane set theory. The paper also uses forcing over non-standard models to obtain new independence results for these weak systems, and uses non-standard models again to prove the unexpected result that the axiom of constructibility, when added to the axiomatisation KPP of Friedman's theory of power admissibility, proves the consistency of KPP.)
[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
(A companion paper, exploring the weakness of Zermelo's
original system for recursive constructions. 7 pp of A4 plain TeX.)
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.
(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.)
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.)
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.
(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.
(derives all the axioms of ZF from a scheme that is apparently
weaker than either the collection or the replacement scheme.)
Unordered pairs in the set theory of Bourbaki 1949
(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.)
Epireflections and supercompact cardinals
Journal of Pure and Applied Algebra 213 (2009) 1208-1215.
(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.)
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.)
Work in progress on RUDIMENTARY RECURSION, PROVIDENT SETS AND FORCING:
Rudimentary recursion and provident sets
(with Nathan Bowler) (submitted)
(We introduce the collections of rudimentarily recursive and gentle 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
.pdf (accepted; revision in progress)
(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.)
Delays, recurrence and ordinals
Proceedings of the London Mathematical Society, (3) 82
(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. )
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.)
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
Choosing an attacker by a local derivation
Carolinae - Math. et Phys., 45 (2004) 59--65.
A scenario for tranferring high scores
Carolinae - Math. et Phys., 45 (2004) 67--73.
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.)
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 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.)
Further remarks on Bourbaki
(A reply to criticism by Professor Sanford L. Segal of the above essay.)
What is Mac Lane missing ?
(a comment on the foundational stance of Saunders Mac Lane; MR 94g:03010;
published with a reply "Is Mathias an ontologist ?" 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.)
Here is a scan of both articles:
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)
Hilbert, Bourbaki and the scorning of logic .ps.pdf
(Since 1999, exposition of formal logic has been banned from the examination for the French Certificat d'Aptitude required for would-be school teachers of 12--15 year olds. This essay traces the grounds for the ban back to the difficulties encountered by the Bourbaki group in their chosen treatment of logic. 60 single-spaced ten-point A4 pages.)
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
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.)
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)
A list of the author's
other mathematical publications is available, and an earlier version of this whole web-site
The author is the Directeur Scientifique of the research group
at the Université de la Réunion.
Reports before truncation:
Reports after truncation: