[Todos CMAT] A cardinal's heresy in classical realizability

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Lun Jul 29 15:36:04 UYT 2013


La charla de Alexandre Miquel sera el miercoles 31 de julio a las 14.30 hs 
en el CMAT. Sala a confirmar, pero probablemente en alguna de las salas de 
seminarios alli disponibles.

Estan todos cordialmente invitados.


A cardinal's heresy in classical realizability

In 2011, Krivine showed how classical realizability can be used to
provide new models of ZF + DC (where DC is the axiom of Dependent
Choices).  In particular, he studied a particular model - the model of
threads - in which infinite subsets of the real line (R) have
cardinality properties that are definitely incompatible with the
(full) axiom of choice and the continuum hypothesis, such as:
1. There is an infinite subset S of R such that SxS is not equipotent
    with S
2. There are two infinite subsets S_1,S_2 of R such that there is no
    surjection from S_1 onto S_2 (and thus no injection from S_2 into
    S_1) and conversely.
3. There is an infinite sequence (S_q) of infinite subsets of R
    indexed by Q (the set of rational numbers) such that:
    - The sequence is increasing w.r.t. inclusion
    - The sequence is strictly increasing w.r.t. cardinality (so that
      the ordering of cardinals is no more well-founded) In this talk, I 
will present these results by rephrasing the entire construction within 
classical second-order arithmetic, to make the underlying ideas and 
techniques more apparent. 

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