[FOM] Restricted powersets for all sets

[Colin McLarty]

Andreas Blass has shown (on mathoverflow) that ZF[1] 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[1] 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[1] probably proves the constructibles
satisfy well ordering.  But I have not checked that tonight. My
application of this result actually uses global choice anyway.

Colin