[Todos CMAT] A cardinal's heresy in classical realizability
mauricio en cmat.edu.uy
mauricio en cmat.edu.uy
Lun Jul 29 11:12:32 UYT 2013
Estimados,
Esta semana, Alexandre Miquel nos propone una charla sobre Realizabilidad
Clasica y Teoria de Conjuntos. La fecha es a confirmar, dependiendo de la
agenda de los miembros del grupo de Logica, pero la confirmaremos a la
brevedad.
Les envio el Abstract en el cuerpo del mensaje. Cordiales saludos,
MAURICIO.
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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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