[FOM] Restricted powersets for all sets
colin.mclarty at case.edu
Mon Jul 30 02:18:01 EDT 2012
Andreas Blass has shown (on mathoverflow) that ZF indeed does not
prove every set has a set of all its countable subsets. His proof
shows what are the key considerations here.
So now, as to consistency strength, I can prove ZF plus "every set
can be well ordered" has an inner model where every set has a set of
all its countable subsets. It is just the standard inner model of
constructible sets, restricted to sets constructed below aleph_omega.
I believe (with encouragement from Andreas) the assumption of well
ordering is otiose because ZF probably proves the constructibles
satisfy well ordering. But I have not checked that tonight. My
application of this result actually uses global choice anyway.
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