On Sunday, November 10, 2013 4:30:56 PM UTC, David Hartley wrote: > In message <BWpz$uCS15fSFw6Q@212648.invalid>, David Hartley > > <firstname.lastname@example.org> writes > > >As far as I know (which isn't far) Ramsey's theorem only applies to > > >countable sets so Rado's proof will only work with countable A. > > > > That was a bit silly of me. For any A, if Ramsey's theorem applies to an > > infinite subset B of A then it applies to A. So, if you're worrying > > about choice, it is certainly true for at least any set with a > > denumerable subset. > > > > The statement and proof of Ramsey's theorem in the Erdos-Rado paper > > apply only to sets of natural numbers. They can easily be extended to > > any countably infinite set but not obviously to an infinite set with no > > denumerable subset. So if you want the Canonical theorem to apply to > > all infinite sets, neither proof works without at least a weak form of > > choice. > > > > I suspect that Rado's proof can be modified to show that if Ramsey's > > theorem extended to all infinite sets holds in ZF, then so does the > > extended canonical theorem, but I haven't checked it through carefully. >
Agreed with everything that you've said in this subthread. However, I have discovered a problem among these papers (aside from problems in my own understanding). Ramsey's original paper is at the URL: www.cs.umd.edu/~gasarch/BLOGPAPERS/ramseyorig.pdf
The underlying set for that paper is "an infinite class" with no suggestion whatsoever that this set is countable. Erdos-Rado then make at least two errors in their exposition. 1) They say that they are proving Ramsey's original theorem but they aren't because theirs is weaker, in assuming a countable underlying set. 2) This is much more serious. They praise their own proof of Ramsey's theorem as being "choice-free" and contrast this with Ramsey's proof. This is misleading because the presentation can only be choice-free if the underlying set is countable.
I don't think a contemporary referee would be happy with the Erdos-Rado statements about how their proof is choice-free.