In article <email@example.com>, "dilettante" <firstname.lastname@example.org> writes: >"Frederick Williams" <email@example.com> wrote in message news:503BDB35.E2CE8638@btinternet.com... >> dilettante wrote:
>>> [...] I suppose it isn't known whether positing the >>> existence of the power set of the reals leads to a contradiction. Surely >>> also taking the existence of this set as an axiom doesn't do anything to >>> resolve CH (of course, as far as anyone knows - if it leads to a >>> contradiction it resolves everything, in a sense)? (Yes, that's a >>> question) >> >> It is known that if the axioms of set theory are consistent then they >> remain so with the addition of either CH (or even GCH) or not-CH. > > But what do the axioms of set theory have to say about the power set of >the reals? Is that a set under the axioms, not a set, or are the axioms >agnostic on the matter?
One of the axioms of ZF is the Power Set Axiom, which can be loosely stated as: if A is a set, P(A) is a set. So your question may be reduced to "are the reals a set under the axioms of ZF?"
(If you were referring to other axioms than ZF, please ignore this post.)
-- Michael F. Stemper #include <Standard_Disclaimer> Life's too important to take seriously.