FOM: Unscientific survey
jlh at math.appstate.edu
Thu May 31 12:21:51 EDT 2001
> 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?
> Thomas Forster
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.
Jeff Hirst jlh at math.appstate.edu
Professor of Mathematics
Appalachian State University, Boone, NC 28608
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