[FOM] BETA(N)

[Nik Weaver]

Most of Harvey Friedman's latest post in this thread is simply a
restatement of thing's he's said before, together with repeated
assertions that he's right and I'm wrong.  So I won't respond to
these comments in detail.  E.g.,

> > That's because impredicative mathematics is justified on
> > platonic grounds which do not hold up under scrutiny.
>
> It holds up quite well under scrutiny. Do you see any problem
> with it?
>
> The only problem you have mentioned in all of your postings
> is simply that it isn't predicative.

Not to put to fine a point on it, I think I have mentioned
other problems (and I never identified "simply that it isn't
predicative" as a problem in itself).

There are two comments in the message that require special
comment, though.  The first is the outright falsehood

> You have already hinted at an expansion of predicativity via the
>
> **Pi11 comprehension axiom scheme**
>
> which has been roundly rejected as predicative by the community of
> predicativists.

--- of course I did no such thing, and I have no idea where this
comment comes from.  I discuss questions like this in detail in
my Gamma_0 paper and clearly explain why Pi-1-1 comprehension is
impredicative.

The other issue deals with Friedman's earlier claim that "in
predicativity, beta(N) does not have any nontrivial elements."
I pointed out that beta N does not even exist predicatively
because we treat ultrafilters over N as proper classes, but
on the other hand there do predicatively exist nontrivial
ultrafilters.

Friedman now says

> Beta(N) is therefore a space of proper classes.

So apparently he accepts the first point.  As to the second, he
challenges me:

> Give us some predicative examples of nontrivial linear functionals.
> If the identification with ultrafilters is appropriate, then you
> would be giving an example of a nonprincipal ultrafilter on N. How
> are you going to do that?

It's trivial in J_2 since there exists a universal well-ordering.
Do you need me to write an explicit formula?

Actually you come close to answering your question yourself:

> For example, if you take any standard system for predicativity
> then it will not even be provable that
>
> *some proper class is a nonpincipal ultrafilter*
>
> In order to derive this statement, you have to add some assertion
> in the predicative theory to the effect that
>
> *the sets of integers are the sets of integers in some L(lambda)*
>
> and even this might not work.

This would be more meaningful if you identified a "standard system
for predicativity" to which the comment is supposed to apply.  But
in any case the predicative legitimacy of an axiom asserting that all
sets are constructible has been universally accepted at least since
Wang's work in the 1950's.  So the predicative existence, as proper
classes, of nontrivial ultrafilters over N is not controversial.

Nik