LIST B: FOUR PAPERS ON SYSTEMS WITH THE POWER SET AXIOM
[B1] The Strength of Mac Lane Set Theory
Annals of Pure and Applied Logic, 110
(2001) 107--234.
.ps.dvi
(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.)
[B2] Slim models of Zermelo Set Theory
Journal of Symbolic Logic 66
(2001) 487--496..ps
.dvi
(A companion paper, exploring the weakness of Zermelo's
original system for recursive constructions.)
[B3]
Unordered pairs in the set theory of Bourbaki 1949
Archiv der Mathematik 94 (2010) 1--10;
MR 2581327.
.ps
.dvi
(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.)
[B4] 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
(Over certain transitive models of Z, the usual treatment of forcing goes awry.
But the provident closure of any such model is a provident model of Z, over which, following [C3],
forcing works well. In [B1] a process is described of passing from a transitive model of Z + TCo to what is here called its lune, which is a larger model of Z + KP. Theorem: Over a provident model of Z, the two operations of forming lunes and set-generic extensions commute. Corresponding results hold for transitive models of Mac Lane set theory M + TCo)}
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