[FOM] hereditarily countable sets and reals
Andreas Blass
ablass at umich.edu
Wed Jan 13 12:39:13 EST 2010
Thomas Forster asked whether one can prove, without the axiom of
choice, that the set H_{aleph_1} of hereditarily countable sets and
the set R of reals have the same size. The answer is negative.
H_{aleph_1} includes the set of countable ordinals, whose cardinality
is aleph_1. But in the absence of the axiom of choice, there might
not be any subset of R of cardinality aleph_1. For example, there is
no such set in Solovay's model for "all sets of reals are Lebesgue
measurable." (In fact, if I remember correctly, it is a theorem of
Shelah that the same holds for all models of "all sets of reals are
Lebesgue measurable," not just Solovay's model.)
Andreas Blass
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