From: weemba@sagi.wistar.upenn.edu (Matthew P Wiener)
Newsgroups: sci.math
Subject: Re: Aleph Question
Date: 14 Oct 1998 20:20:34 GMT

In article <6vtob9$6ok@mcmail.cis.McMaster.CA>, kovarik@mcmail (Zdislav V. Kovarik) writes:
>In article <36210C35.49C59A9B@rocketmail.com>,
>Peter Ammon  <Peter_Ammon@rocketmail.com> wrote:
>:Is Aleph_2 or Aleph_anything greater than 1 interesting at all?

>:-Peter

>They pop up whether they are "interesting" or not:

>Assuming GCH, aleph_2 is the cardinality of the space of bounded real
>functions on [0,1], for example. This Banach (Banachable :-?) space is
>useful in modern analysis, when you prove Riesz representation theorem of
>the dual space to C([0,1]). 

(I use w for omega=omega_0=aleph_0.)

The cardinality of the Stone-Cech compactification of N is 2^2^w,
which is aleph_2 assuming GCH.

The cardinality of Mary Ellen Rudin's Dowker space (normal Hausdorff,
but not countably paracompact [equivalently, its product with [0,1]
is not normal]) is aleph_w ^ aleph_0, which is aleph_(w+1) assuming
GCH.

For almost thirty years, the question of whether there was a smaller
one has been a fascinating set theoretical topological problem.  All
sorts of c or aleph_1 sized Dowker spaces were known (going back to
Rudin's work in the 50s).

Rather recently, Balogh constructed a c-sized Dowker space in ZFC
alone.  Shelah and Komjath then constructed a aleph_(w+1) sized
Dowker space in ZFC, essentially taking a pcf-ied subspace of the
Rudin ZFC example.  Theirs is the smallest sized known in ZFC.

aleph_w is of special interest due to a rather bizarre result of
Shelah, also from pcf theory.  Assume k<aleph_w implies 2^k<aleph_w.
Then 2^aleph_w<aleph_aleph_4.  Assuming large cardinals, models are
known with 2^aleph_w=aleph_a, for most a<aleph_1.  Shelah thinks his
result is optimal.

And speaking of large cardinals, many of them are of interest.
-- 
-Matthew P Wiener (weemba@sagi.wistar.upenn.edu)