venerdi' 11-03-2005 (16:00) - Sala Riunioni Joan Bagaria (ICREA Barcelona) : Natural Axioms of Set Theory and the Continuum.
As is well-known, Cantor's continuum problem, namely, what is the cardinality of the set of real numbers, is independent of the usual ZFC axioms of Set Theory. K. Goedel suggested that new natural axioms should be found that would settle the problem and hinted at large-cardinal axioms as such. However, shortly after the invention of forcing, it was shown that the problem remains independent even if one adds to the standard ZFC system of Set Theory the usual axioms of large-cardinals, like the existence of measurable cardinals, or even supercompact cardinals, provided, of course, that these axioms are consistent. While numerous axioms have been proposed that settle the problem--although not always in the same way--from the Axiom of Constructibility to strong combinatorial axioms like the Proper Forcing Axiom or Martin's Maximum, none of them so far has been recognized as a natural axiom and been accepted as an appropriate solution to the continuum problem. In this talk we will discuss some heuristic principles, which might be regarded as Meta-Axioms of Set Theory, that provide a criterion for assessing the naturalness of the set-theoretic axioms. Under this criterion we then evaluate several kinds of axioms, with a special emphasis on a class of recently introduced set-theoretic principles, known as axioms of generic absoluteness, for which we can reasonably argue that they constitute very natural axioms of Set Theory and which settle Cantor's continuum problem.
Giulia Curciarello Segreteria Didattica tel: 050-2213219 e-mail curciare@dm.unipi.it _______________________________________________ Settimanale mailing list Settimanale@mail.dm.unipi.it https://mail.dm.unipi.it/mailman/listinfo/settimanale