[CvGmt News] [Notiziario] [Settimanale] cambio di orario seminario- Joan Bagaria

[Giulia Curciarello]

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 at dm.unipi.it
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