Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: Surprise at my failure to resolve an issue in an elementary paper by Rado
Replies: 44   Last Post: Nov 10, 2013 12:23 PM

 Messages: [ Previous | Next ]
 Paul Posts: 780 Registered: 7/12/10
Re: Possible major blunder in Rado's version of Canonical Ramsey
Theorem that goes far beyond omitting proof steps

Posted: Nov 10, 2013 12:23 PM

On Sunday, November 10, 2013 4:30:56 PM UTC, David Hartley wrote:
> In message <BWpz\$uCS15fSFw6Q@212648.invalid>, David Hartley
>
> <me9@privacy.net> 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.

Paul Epstein

Date Subject Author
11/3/13 Paul
11/3/13 David Hartley
11/3/13 fom
11/3/13 fom
11/3/13 fom
11/4/13 fom
11/4/13 Paul
11/4/13 Paul
11/4/13 Peter Percival
11/4/13 David Hartley
11/4/13 Paul
11/4/13 David Hartley
11/4/13 Paul
11/4/13 David Hartley
11/4/13 Paul
11/5/13 Paul
11/5/13 David Hartley
11/5/13 Paul
11/5/13 David Hartley
11/5/13 Paul
11/6/13 Paul
11/6/13 Paul
11/7/13 Paul
11/7/13 David Hartley
11/7/13 Paul
11/7/13 David Hartley
11/7/13 Paul
11/7/13 David Hartley
11/7/13 David Hartley
11/7/13 Paul
11/7/13 David Hartley
11/8/13 Paul
11/8/13 David Hartley
11/7/13 Paul
11/7/13 fom
11/8/13 Paul
11/8/13 David Hartley
11/10/13 Paul
11/10/13 David Hartley
11/10/13 Paul
11/10/13 David Hartley
11/10/13 David Hartley
11/10/13 Paul
11/4/13 Paul
11/4/13 Peter Percival