> On Sat, 27 Oct 2012 22:58:21 +0100, Frederick Williams > <email@example.com> wrote: > > > JRStern wrote: > >> > >> Are there any such published? > >> > >> I can see in the archives here it's a common topic, and I have my > own >> crackpot theories which certainly overlap a lot of the more > popular >> objections. > >> > >> I don't want to prove or assert or reject any statement about the > >> countability of reals, I just want to consider the validity of the > >> diagonalization argument. > >> > >> Has anybody put that out in a refereed journal or a respectable > >> publisher? Even if it's just a prettier rejection of crank > theories, >> it would seem worthwhile. > > > > In his lifetime Cantor was opposed by some very competent > > mathematics: Brouwer, K\"onig, Kronecker, Poincar\'e, Weyl. > > Yes, and I agree with most of them, only those are > intuitionist/constructivist, and they rejected large parts or all of > Cantor's cardinals, reasoning, and I suppose necessarily reject ZFC. > > I want to see if within ZFC, the diagonal argument itself is valid. >
Yes. It is easily recast in the language of ZFC. Cantor in fact proved that Card(Powerset(S)) > Card(S) for all sets S, which is a generalisation of the proof that c is uncountable.
> I suppose at worst it could be made an optional axiom, or some > statement of principle could be added as an additional axiom to allow > it to be true. > > J.