From: Fred Galvin Subject: Re: Size of 2^Aleph0 Date: Mon, 10 Jan 2000 02:10:02 -0600 Newsgroups: sci.math Summary: [missing] On Mon, 10 Jan 2000, Jonathan Hoyle wrote: > As most readers on sci.math know, the Continuum Hypothesis > (2^Aleph0 = Aleph1) is independent of the axioms of set theories > ZFC and ZF. Apparently, it is consistent to take as an axiom > 2^Aleph0 = AlephN for almost any ordinal N. However, recently I > read that there was a proof that N could not be omega (the first > infinite ordinal). > > Are there any other restrictions on N? Can N be any other limit > ordinal? N can't be 0, and it can't be an ordinal of cofinality omega; it can be anything else. > How about N = 2^Aleph-omega? That is possible. > If we do not assume AC, must 2^Aleph0 be an aleph at all? That's tantamount to asking, if we do not assume AC, must there be a well-ordering of the set of all real numbers? The answer is no. > I am told that currently there is no upper limit the size of > 2^Aleph0 (it could be the first hyper-inaccessible cardinal, the > first Mahlo cardinal, etc). Let's be careful: there's "strongly inaccessible" and "weakly inaccessible", and the default meaning of "inaccessible" is "strongly inaccessible". A weakly inaccessible cardinal is a regular limit cardinal. A (strongly) inaccessible cardinal is a regular *strong* limit cardinal; m is a strong limit cardinal if n < m implies 2^n < m. Of course, 2^Aleph0 is not *strongly* inaccessible; however, it can be weakly inaccessible, weakly hyperinaccessible (whatever that means, I forget), weakly Mahlo (whatever that means), etc. If it's consistent to assume that there is a measurable cardinal greater than omega, then it's consistent to assume that 2^Aleph0 is a "real-valued measurable cardinal" (RVMC), and if 2^Aleph0 is a RVMC, then it is very high in the scale of weakly inaccessible cardinals. By the way, if 2^Aleph0 is a RVMC, it follows that 2^m = 2^Aleph0 for every infinite cardinal m < 2^Aleph0; it particular, 2^Aleph-omega = 2^Aleph0, and so we have 2^Aleph0 = AlephN where N = 2^Aleph0 = 2^Aleph-omega.