[FOM] Weaver's error?

[Harvey Friedman]

On 4/9/06 3:29 AM, "Nik Weaver" <nweaver at math.wustl.edu> wrote:

> At least two messages have now been posted which say that my
> objections are unconvincing, but don't say why.

I have already said why, and I will repeat it again, hopefully being
clearer. Also, I comment on your second "objection" to Feferman/Schutte at
the end of this message.

There is no single philosophical notion being analyzed in a fundamental way.

Of course, "predicativity" does carry some philosophical meaning, although
not at all definite or unique.

The Feferman/Schutte formalism is

1. Totally consistent with the normal non-unique idea of "predicativity".
2. Is an interesting place in the robust hierarchy of logical strengths
ranging from EFA to j:V into V.

>From the viewpoint of actual foundations of actual mathematics, already it
doesn't make much difference if the use Feferman/Schutte or my strict
predicativity of http://www.cs.nyu.edu/pipermail/fom/2006-April/010360.html

Kruskal's theorem is so off the charts beyond anything that a normal
mathematician would regard as "predicative" - after hearing the brief
informal explanations of, say, Russell (who also authored the axiom of
reducibility!) and Poincare and Weyl and others - that any objection to
calling KT impredicative simply amounts to a promotion of some special
EXTREMELY LIBERAL OUTER LIMIT controversial interpretation, among many, of
"predicativity". It is totally misleading, and represents a fundamental
misunderstanding of the circumstances surrounding "predicativity".

Here are some more detailed comments.

1. The whole idea of taking well-orderings and what you can do with them
(recursions and transfinite inductions), as philosophically fundamental, is,
in my opinion, wrong headed. In a truly convincing philosophical treatment,
one must start from far more fundamental ideas - especially comprehension
itself. 

2. There hasn't been any convincing philosophically fundamental analysis of
"predicativity". I do not preclude the possibility that there MIGHT be such
a thing in the following sense. There MIGHT be some particular
philosophically fundamental explication of a sense of "predicativity" which
goes beyond the mere WORD, which DOES lend itself to an analysis together
with an entirely convincing completeness theorem.

3. I am an optimist. I still think hard from time to time on how to do this
for effective computability, and have made some FOM postings on an approach
to doing this, in the past.

4. In any case, right now I view the Feferman/Schutte analysis as the
delineation of a very interesting stopping place that is wholly consistent
with the informal idea of "predicativity".

5. So Weaver's "criticisms" amount to nothing more than ARGUING with a
stopping place! How can you argue with a stopping place? Well, you can say
that the stopping place is NOT INTERESTING. This, Weaver isn't even claiming
to do.

6. I will comment directly on Weaver's second objection, which takes the
general form of: 

1) "predicativists" X act under a particular rule: from A derive A*.

Therefore 

2) "predicativists" X accept the implication A implies A*.

Weaver says that according to Feferman, 1) holds. Weaver then says that
Feferman must admit 2) holds. But then Feferman must admit that
predicativists accept that Gamma_0 is well ordered. So Gamma_0 is too small,
and the Feferman/Schutte analysis is wrong.

7. This objection seems to be based on a common error. Weaver's "inference"
is not valid. It would be ARGUABLY valid if 1) is strengthened to

1') "predicativists" X ACCEPT THE VALIDITY of a particular rule: from A
derive A*.

Therefore

2) "predicativists" X accept the implication: A implies A*.

Now Feferman accepts the validity of a particular rule: from A derive A*.

Or, put more appropriately, Feferman accepts the implication

3) if "predicativists" X accept A then "predicativists" X accept A*".

However, Feferman, in his analysis of predicativity, is NOT CLAIMING to act
entirely as a predicativist himself. He is only claiming to analyze the
thinking of a "predicativist". (I'm not sure he would put it this way
exactly, but perhaps this is close enough for present purposes).

So Feferman would not put forth 1'), but only 1).

8. This kind of objection of Weaver's can surface against any proposed
closed end analysis of anything, and is generally disregarded along the
lines I indicated. Otherwise, as Feferman points out in his letter to Weaver
on Weaver's website, one could automatically object to any formal analysis
of a very general kind.

9. The portion of Weaver's message that amounts to this error is seen below:

> ... for each a he has some way to make the deduction
> 
> (*)   from  I(a)  and  Prov_{S_a}(A(n)),   infer   A(n)
> 
> for any formula A, where I(a) formalizes the assertion that a is an
> ordinal notation (supporting transfinite induction for sets).
> 
> Shouldn't he then accept the assertion
> 
> (**)   (forall a)(forall n)[I(a) and Prov_{S_a}(A_n)  -->  A(n)]
> 
> for any formula A?
> 
> It is easy to see that the second assertion implies I(a) where a
> is a notation for Gamma_0.  So we somehow have to accept every
> instance of the rule (*) but not the general implication (**).
> Why?

Harvey Friedman