[FOM] BETA(N)

[Nik Weaver]

Harvey Friedman:

> 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?

Huh?  That's exactly backwards.  Predicatively, ultrafilters
over N are treated as proper classes.  So beta N does not exist
as an object but individual (nontrivial) ultrafilters do.

A quick way to see this is by identifying ultrafilters over N
with homomorphisms from l^infinity into the scalars.  As I
explained in a previous post, the dual of l^infinity does not
exist predicatively but individual linear functionals do exist.

> 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.

See the previous message about accord with mathematical practice.
To the extent that beta N really has been centrally used in core
mathematics my point about "exact fit" is weakened.  To the extent
that uses of beta N are actually inessential my point is strengthened.

Most core mathematicians I know tend to regard beta N as a highly
pathological object that is better avoided.

Nik