[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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