It's not entirely correct to say that the statement of CH doesn't make sense w/o AC. In fact, in many ways it's more interesting that way. You say without AC we can't compare the sizes of infinite sets. More accurately, in ZFC, cardinality is a (class) linear or total order relation. In ZF it is a (class) partial order relation. Thus we can compare sizes, but sometimes the answer is "incomparable." Consider the following 2 statements: [Here c = 2^aleph_0 = continuum] (1) There is no cardinal k : aleph_0 < k < c (2) c = aleph_1 In ZFC, these statements are equivalent and correspond to CH. In ZF, they are not equivalent. (2) => (1) but (1) !=> (2). Also (2) => the reals can be well ordered. (1) does not. Another relation I found even more surprising related to the Generalized CH: (1) For every infinite cardinal k, there is no cardinal j: k < j < 2^k (2) For every ordinal a, 2^aleph_a = aleph_{a+1} In ZF, (1) => 2, (2) !=> (1), (1) => AC, (2) does not. ------ Original Message ------ Received: Sun, 09 Dec 2012 08:22:41 AM PST From: "Adam P. Goucher" <apgoucher@gmx.com> To: "Warren Smith" <warren.wds@gmail.com>,math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] colouring of R^3 ...

NB: I've liberally assumed the Axiom of Choice everywhere, but that's okay because the statement of the continuum hypothesis doesn't even make sense without it (without AC, we can't compare the sizes of infinite sets).

...