FOM: Unscientific survey

[Jeff Hirst]

> I would like to get a feel for what the
>received view is on the following question:
>
>  Is Ramsey's theorem a theorem in cardinal arithmetic?
>
>     thanks
>
>       Thomas Forster


Hi Thomas-

I'm going to assume that what you're thinking of as
Ramsey's theorem is the usual infinite version of
Ramsey's theorem.  (The set theoretic notation is
\forall n \forall k \omega -> (\omega)^n_k and in
the arithmetic literature, we ususally write RT or
\forall n \forall k RT(n,k).)

Yes, I think this is a theorem in cardinal arithmetic.
I also think it's a theorem of graph theory (in the
obvious fashion for n=2, and in less obvious ways
for higher exponents.)  It's also a theorem of
second order arithmetic, but I don't think of it as
a number theoretical statement, since \omega can be
replaced by any countable set.

I've always thought it was very odd that the arrow
notation is written with \omega rather than \aleph_0,
since we are clearly thinking of \omega as a cardinal
in this setting.  I'd be interested in hearing about the
history of the arrow notation.

My other guess is that saying that Ramsey's theorem is
a theorem in cardinal arithmetic is somehow philosophically
loaded, and might have consequences that I would find
uncomfortable.  I hope you'll fill me in on those.

Thanks,

-Jeff

-- 
Jeff Hirst   jlh at math.appstate.edu
Professor of Mathematics
Appalachian State University, Boone, NC  28608
vox:828-262-2861    fax:828-265-8617