[FOM] BETA(N)

[Harvey Friedman]

On 2/22/06 4:32 PM, "griesmer at math.ohio-state.edu"
<griesmer at math.ohio-state.edu> wrote:

> Beta(N) is
> an important object in functional analysis, topological dynamics, and
> Ramsey theory; its structure has far-reaching combinatorial consequences.
> For instance, one may use information about the topological and algebraic
> structure of beta(N) to conclude the following:  If N is partitioned into
> finitely many classes, one cell of the partition will contain each of the
> following:
> 
> (i)   Arithmetic progressions of every finite length.
> (ii)  Geometric progressions of every finite length.
> (iii) An infinite set A and every finite sum with distinct summands from A.
> (iv)  An infinite set B and every finite product with distinct factors
> from B.
> 
> (Neil Hindman has shown that we cannot necessarily take A=B above.)
> 
Weaver responded to Griesmer's posting, but I don't think that he addressed
the following issue.

beta(N) is the space of ultrafilters on N. Hence in predicativity, beta(N)
does not have any nontrivial elements.

So how is beta(N) to be handled in predicativity?

Even if beta(N) can be removed from these very interesting applications
i)-iv), the FACT is that the relevant mathematicians CHOSE to use beta(N) in
their proofs. This was of their own free will. Weaver needs to comment on
that.

Certainly by work of Gowers, (i) has been proved very explicitly, with good
bounds as to how far up you have to look. Thus the beta(N) is eliminable
here.

What about ii-iv? Have they been proved with similar explicitness?

What would be involved in trying to eliminate the use of beta(N) in ii)-iv)?
Is there a down to earth treatment?

Harvey Friedman