[FOM] BETA(N)

[Harvey Friedman]

On 2/24/06 11:23 PM, "Nik Weaver" <nweaver at math.wustl.edu> wrote:

> 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.

This is not a correct reflection on my latest post.

1. I explicitly asked you to compare which is closer to the core
mathematical practice of the French algebraic geometry school with its
Grothendieck universes and topoi - set theory or predicativity? You have not
responded to this point.

2. I also discussed the prevalent use of impredicative reasoning combined
with Zorn's Lemma in core countable algebra. You have not responded to this
issue. Are you waiting for me to supply more information before responding,
or do you have some other reason for not responding?

3. I also suggested that as far as I can tell, ACA0 already does the jobs
that you are talking about in your predicative interpretation of functional
analysis. Also much weaker systems probably suffice. So again, it becomes
clear that predicativity is not any particularly distinguished match for
core or normal mathematical practice. You have not responded to this point.

4. You also have not addressed the question of why ZFC has gained such
overwhelming acceptance in the mathematical community for over 80 years.

5. You have not addressed the issue that I raised that referring to
impredicativity as "having no clear philosophical basis" is a misuse of the
word "philosophy". 

6. You have not addressed the issue of the nature of your condemnation of
impredicative mathematical activity is, including whether it is appropriate
to continue to teach and publish impredicative mathematics, and whether it
should be identified in publications as "having no clear philosophical
basis". 

7. You have not addressed the issue of how you are going to treat basic work
done on functionals between beta spaces, predicatively. I mentioned, e.g.,

Andreas Blass (and coauthor) Finite Preimages Under the Natural Map from
beta(N x N) to (beta N)x(beta N), joint with Gugu Moche (Topology
Proceedings 26 (2001-2002) 407-432)

available from http://www.math.lsa.umich.edu/~ablass/set.html

If you have answered any of these seven points, I would appreciate it if you
would, for each of the seven, either

1. Answer them now.
2. Answer them again.
3. Give a pointer to the posting in whichthey were previously answered,
together with a reference to the relevant passages - if those passages are
not apparent.
4. Acknowledge the question and say that you do not have an answer.

I have directly answered every single one of your points, and expect the
same in return. Fair enough?

Friedman wrote:

>> 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).

You have not cleaerly identified any difficulty with impredicativity. I see
nothing mentioned except that it is not predicative.

The closest "problem" that you have identified is that impredicativity
depends on some form of Platonism. All that does is shift the issue to a new
word: Platonism. 

You haven't identified any problem with Platonism, except to say that some
extreme forms of what you call Platonism were shown to be inconsistent
(Russell's paradox).

This extreme form of what you call Platonism is not only grossly outdated,
but was obviously not even close to any form of Platonism that was in
operation in core or normal mathematics at any time in history. That's why
Russell's paradox had no effect on normal or core mathematical practice.

At the same time, basic impredicativity is all over core and normal
mathematics and generally accepted by mathematicians as a fundamental tool
of the trade - a basic proof technique.

Impredicativity in normal or core mathematics stops well short of any kind
of extreme Platonism that has even the slightest hint of a problem. It stops
well short of the very focused and basic form of what you call Platonism,
that is embodied in Z_2.

In particular, you have given not even the slightest hint of any problem,
conceptual or mathematical, with Z_2, ZC, or that Gold Standard, ZFC.
 
Friedman wrote:

>> 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 wrote this because you mentioned that it was open whether or not the graph
minor theorem was predicatively provable, in an earlier posting of yours.
That's how I got the impression.

GMT cannot be proved in Pi11-CA0 and furthermore it proves the consistency
of Pi11-CA0. Prizes have been awarded by the AMS and other organizations
largely based on GMT. Do you wish to dismiss GMT as not "core" or not
"normal" mathematics? GMT is certainly yet another major challenge to
predicativists - one that you have not acknowledged as far as I recall.

I'm glad to hear that you have chosen to answer this question about Pi11-CA0
explicitly - even though I didn't ask it explicitly. It would be useful if
you would answer the other questions that I have asked explicitly.

> The other issue deals with Friedman's earlier claim that "in
> predicativity, beta(N) does not have any nontrivial elements."

>> 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?

I didn't make my points clear.

1. The technical construction you mention is based on a system for
predicative mathematics that is not appropriate for use in mathematical
practice. Mathematicians are not going to want to use an axiom like "all
sets of natural numbers lie in J_2". Mathematicians are going to want to
have a proof that there exists nonprincipal ultrafilters. And that proof
cannot use a lot of technical mumbo jumbo.

2. The set theoretic systems do not have this problem, such as Z, ZC, ZF,
ZFC, etcetera. In ZC, ZFC one formalizes the usual mathematical proof that
there is a nonprincipal ultrafilter. Here, as mathematicians do proofs, they
can actually quote the axioms that they use, and those axioms will be of a
simple nature.  

3. A system I keep mentioning that seems to do essentially all of the work
that predicativity does for mathematical practice, is ACA0. On the other
hand, ACA0, augmented with proper classes, is certainly not going to prove
that there exists a nonprincipal ultrafilter as proper class.

4. Another point is this. You don't really have an ultrafilter on N, even as
a proper class, unless the ultrafilter interacts decently with the elements
of P(N). By this I mean the following.

Suppose that you have an infinite sequence x1,x2,... of subsets of N, which
exists as a single subset of NxN. The ultrafilter can be applied to these
x's to produce a sequence of bits. You want that sequence of bits to form a
legitimate subset of N.

Why do you want this? Because it amounts to, say arithmetical comprehension
with the ultrafilter as a parameter, and I don't if you can do anything
interesting or useful in beta(N) theory without such a principal.

5. If you go this technical route of having a system for predicativity based
on technical mumbo jumbo, then it seems like you can take care of point 4
above with some extra care in the construction.
> 
> 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.

How are you going to state in the system that all sets are constructible?
First of all, it is mathematically obnoxious, although I managed to make it
better than expected in the context of set theory in an earlier posting this
month. Secondly, the only way to state it, generally, is to use the concept
of *well ordering* which is rather atrocious predicatively. Also the
assertion

every well ordering can have recursion performed on it

is regarded by (almost all) predicativists as not predicatively provable or
acceptable.

So the obvious choice, if one wishes to incorporate "all sets are
constructible" in the formal system is to actually mention specifically the
ordinal of the universe. But this is technical and arguably inappropriate
for a formalization of mathematical practice.

Harvey Friedman