In article <email@example.com>, firstname.lastname@example.org wrote:
> Let us simply restrict ourselves to the cases where one makes no > assumption regarding CH, and where one asserts CH. > > How is this conceptually different for you than considering on the one > hand the usual axioms of group theory, and on the other assuming in > addition commutativity, so that all groups are Abelian?
For me, it's very different, as follows. If I change the definition of a group to include commutativity, S_3 still exists; it's not a group any more, but it is a mathematical object about which I can prove theorems. If I change the definition, and you don't, you and I still have the same universe of mathematical objects - we just don't use the same words to describe them.
But if you accept CH, and I don't, then I have sets of real numbers that are intermediate in cardinality between the integers and the reals, and you don't. I can prove theorems about these things, and you can't even see them. That (to me) is a big conceptual difference between assuming commutativity and assuming CH.
-- Gerry Myerson (email@example.com) (i -> u for email)