[FOM] "Refutation" of the Feferman-Schutte analysis of predicativism

[Nik Weaver]

Harvey Friedman accepts my observation that the Feferman-Schutte
analysis presumes that predicativists are capable of accepting
every instance of a deduction rule (*) but cannot accept the
corresponding implication (**).  (See the message
http://www.cs.nyu.edu/pipermail/fom/2006-April/010368.html
for details.)

However, he provides a very inaccurate summary of my argument:

> We now know that Weaver claimed to have "refuted" the
> Feferman/Schutte analysis merely by pointing out that
>
> 1. Weaver believes *).

This is false.  I don't accept (*) as a general principle.  I
certainly never "pointed out" that I believe (*).

More to the point would be whether (*) is predicatively acceptable.
As I've explained several times now, I don't think it is.

> 2. Weaver believes that Feferman/Schutte don't argue properly for *).

True.  Actually, I haven't seen any explicit argument in its favor.
It seems to be taken for granted.

> 3. Weaver believes **).

False.

> 4. Feferman/Schutte deny **).

True.

> 5. Weaver believes that Feferman/Schutte must give explicit
> reasons for explicitly denying **). [NOTE: Weaver believes
> that Feferman/Schutte cannot rely on the fact that there is
> no implication from *) to **). Weaver believes that If they
> don't, then Feferman/Schutte has been REFUTED by Weaver.]

No, I wouldn't say this.  I think proponents of the F-S analysis
definitely have a burden to explain why (*) is predicatively
acceptable, since on its face it is not.  Such explanations can
then be examined to see whether they actually justify (**), which
would lead predicativists beyond Gamma_0.

I have pointed out that it is hard to imagine how one would justify
acceptability of each instance of (*) without simultaneously
justifying (**), and I have analyzed three attempts by Kriesel to
do this, and, I believe, shown them to be erroneous.  Does that
count as a "refutation"?

> If Weaver had instead been claiming all along that
>
> 8. Weaver does not find the Feferman/Schutte analysis of
> predicativity convincing.
>
> then I would not have disagreed with him.

I don't believe this.  Indeed I would say it is contradicted by
the following passage from an earlier message of Friedman,
quoting me:

> > The fact that the F-S analysis has been almost universally
> > accepted for over 40 years indeed troubles me.
>
> It doesn't trouble anyone else that I know for these reasons.
>
> 1. There is no convincing defect.
> 2. Although there are weak points that can be complained about,
> this is true of any (what you call) "foundational stance".

I think Friedman now sees that the application of (*) is a
fundamental defect in the F-S analysis.  In order to accept any
instance of (*) we apparently have to be able to form the set
{a: S_a is sound}, requiring a comprehension axiom well beyond
anything that anyone thinks is predicative.  This criticism is
indeed specific to the case of predicativism and Gamma_0, as I
said earlier (here being quoted by Friedman):

> > You haven't read my criticism.  It is highly specific to the
> > case of predicativism and Gamma_0.
>
> Not really. Tait has said that finitism is encapsulated by PRA.
> But you can see the Ackermann function from below. Tait has a
> defense, and Kreisel can attack again. And so forth. The details
> are always different, but the flavor is always the same.

Now that Friedman is more familiar with my argument, would he
still assert that "the flavor is always the same"?

Friedman now writes:

> It is essential that Weaver change his clear misusage of the word
> "refute" so that the discussion of predicativity between Friedman
> and Weaver can now focus on some matters of import. It hasn't yet,
> and the obstacle has been Weaver's wholly inappropriate use of the
> word "refute".

It seems to me that a more serious obstacle has been Friedman's
willingness to jump to conclusions about the nature of my arguments,
leading him to repeatedly attack positions that I don't actually hold.

As to the appropriateness of the word "refute", I don't think this
is a substantive question.  However, I would not want to overstate
my case or claim that I've done more than I really have.

Here is what I claim to have done (in detail, in my paper
"Predicativity beyond Gamma_0"):

--- made explicit the assumption implicit in the Feferman-Schutte
analysis that predicativists can accept each instance of the
deduction rule (*);

--- noted that the corresponding implication (**) would, if
accepted, lead predicativists beyond Gamma_0;

--- noted that application of (*) on its face requires an
impredicative comprehension axiom, so that (*) is surely not
predicatively acceptable;

--- made a case that any coherent foundational stance that led one
to accept each instance of (*) would also lead one to accept (**);

--- shown that three separate attempts by Kriesel to justify (*)
but not (**) fail.

I leave it to interested readers to decide for themselves whether
the above, if successful, constitutes a "refutation" of the F-S
analysis.

Nik