## Monday, November 05, 2007

### It was a stupid question!!!!!!!!!! or...

On my last blog I asked the following: TRUE OR FALSE:
For every coloring of R (the reals) with a countable number of colors there exists x,y,z,w of the same color such that x+y=z+w.

And I pointed out that when I asked this in seminar I got 5 thought it was TRUE, 4 thought it was FALSE, 5 thought it was a STUPID QUESTION.

The answer is: ITS A STUPID QUESTION. More rigorously the following is true and was proven by Erdos:
The statment above is true iff the Continuum Hypothesis is false. (See this (pdf) or this (ps). for an exposition of the proof.

SO, what to make of this? This is a natural question that is ind of ZFC. How Natural is it? Erdos worked on it, not some logician looking around for a problem to be ind of ZFC.

Does this make us think CH is true or false? Actually, more is known:
Let L(x1,..., xn) be a linear form over the reals (but not x1-x2). If CH is true then there is a coloring of the reals with a countable number of colors such that there is no e1,..., en which are all the same color such that L(e1,..., en)=0. (Exposition of proof in same document linked to above.)
If CH is true then the entire theory of countable colorings and linear forms is known. And boring. If CH is false then much more interesting things happen. Jacob Fox proved the following:

Let STAT(s) be the statement
For every coloring of R with a countable number of colors there exists x1, x2, ..., x{s+3} such that they are all the same color, and x1 + sx2 = x3 + x4 + ... + x{s+3}
THEN STAT(s) is true iff 20 > ℵs

Jacob Fox is also (judging from his resume) not a logician. He is a combinatorist. Actually he's a graduate student so it may be too early to say what he is.

To determine CH should we use its consequences as reasons for or against assuming it? Even if we do, do you want the entire theory to be known and boring? I ask this non-rhetorically. See Opinion 68 of Zeilberg's blog or The papers of Penelope Maddy: believing the axioms I. and believing the axioms II.

1. Someone was telling me at Dagstuhl that Hugh Woodin (who was at MIT for a year when Bill and I were both doing logic) has a new argument that CH "ought to be" false -- in fact that the cardinailty of the continuum "ought to be" exactly aleph-two The reference is Notices of the AMS 48 (2001), (6) 567-576 and (7) 681-690..

2. Are you saying it's a stupid question just because it's independent of ZFC?

There are plenty of natural problems that turn out to be ZFC independent. The Whitehead problem is an excellent example. Of course, their density in mathematics as a whole is quite low, but there are plenty of natural examples out there.

You also seem to think independence results are more worthwhile if they're proved by nonlogicians. Why is the result more or less worthwhile depending on who proved it?

3. (This is Bill Gasarch)

Anon: I thought the number
of natural problems ind
of ZFC was very very small,
but would be happy to hear
Email me privately and
we'll discuss.

Also: good point, its not
WHO proved the result
that matters. What I
was getting at is that
a problem that someone
(probably a logician)
comes up with that nobody
in math ever heard of
before, just to find
something ind (e.g.,
Large Ramsey Theorem)
is not as natural as
a problem that someone
(prob not a logician)
was honestly working on
and found to be ind.
(CH, and the subject
of my post).

4. This comment has been removed by the author.

5. Anonymous 2: You also seem to think independence results are more worthwhile if they're proved by nonlogicians. Why is the result more or less worthwhile depending on who proved it?

Statements in math that are indep of ZFC are (a priori) more likely to be generally interesting if they were first stumbled upon by nonlogicians. It's not that they were made interesting this way; they were interesting enough already, which made them more likely to be found by nonlogicians.

That said, here's a counterexample. Harvey Friedman, a logician, has been developing Boolean relation theory, relating to properties of ordinary mathematical objects like functions on sets of integers. It has natural questions that would interest any combinatoricist, and it has theorems that are provably equivalent to large cardinals. He's written a lot about it on the FOM mailing list. Go to
http://cs.nyu.edu/mailman/listinfo/fom
and search for Boolean Relation Theory.

6. Bill,

Thanks a lot for your interesting post, the write-up and the links.

Wikipedia has a short list of mathematical statements that are independent of ZFC on this page. You may find it interesting.

Saharon Shelah's result to the effect that the Whitehead problem is undecidable in ZFC is discussed in this paper by Eklof. Shelah has also shown that the problem is independent of ZFC+CH.