I decided to cross-post this to sci.logic, as I think it is more up
their alley:
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The Power Set Axiom in ZF defines the collection of subsets of any set
is itself a set.  Unfortunately, it does not define what those subsets
are.  The obvious example is in the case of the set of reals, R.  In
ZFC, the construction of a non-measurable set is a member of P(R).  In
ZF, however, such a collection is not a member of P(R).  Without the
Axiom of Choice, a very large number of such collections will not be
sets at all, and thus must be ignored.

What one would like is to have *any* arbitrary collection of real
numbers to be a member of P(R), not leaving out anything.  Is ZFC strong
enough to support this?  Is the request even consistent?  Or must there
always be a collection of reals which will not be recognized as a set,
given any axiomatic system?

If so, can you give an example of such a collection of reals which would
not fulfill the definition of sethood in ZFC (but perhaps might in
another axiomatic system)?

Jonathan Hoyle