[FOM] on Bas Spitters on "constructive impredicativity?"

[Nik Weaver]

Gabriel Stolzenberg wrote:

> As I recall, part of the story was that Harvey had proved that
> every proof of Kruskal's theorem requires impredicativity.

Dear Gabriel,

I think this claim of Friedman's is essentially meaningless.  It
relies on the Feferman-Schutte analysis which asserts that Gamma_0
is the least predicatively unprovable ordinal.  I have extensively
criticized this analysis in my paper "Predicativity beyond Gamma_0";
I recently posted a brief version of my argument in the message
http://www.cs.nyu.edu/pipermail/fom/2006-March/010245.html

I really don't believe there can be any serious opposition to my
position on this.  The original analysis of Feferman and Schutte
based on Kreisel's idea of autonomous progressions simply conflates
classically equivalent versions of the well-ordering concept which
are not predicatively equivalent.  Feferman recognized this defect
("the well-ordering statement ... on the face of it _only
impredicatively justifies_ the transfinite iteration of accepted
principles up to a", in "A more perspicuous formal system for
predicativity", p. 85; italics in original) and proposed a variety
of analyses based on alternative formal systems, but these are all
clearly impredicative as well.  His most recent analysis, made in
collaboration with Strahm, involves a system that uses self-referential
"schematic" predicate symbols which are obviously impredicative.

At a more basic level, the obvious question about why, assuming a
predicativist could make it up to Gamma_0, he could not go further,
was never satisfactorily answered.  Kreisel addressed this objection
several times but never settled on a workable response.  Again, see
my Gamma_0 paper for a detailed discussion.

If you're interested in these issues, you may want to look at
Feferman's response to my paper, and my answer to him, which are
both posted (with Feferman's permission) together with the paper
itself at
http://www.math.wustl.edu/~nweaver/conceptualism.html

When I asked Friedman directly whether he was willing to assert that
I am wrong about all this, he replied

> I am asserting that the concept of "predicativity" is not
> sufficiently clear to allow anything but a broad range of equally
> legitimate interpretations. So the established convention given
> by Feferman and Schutte will stand for the forseeable future.

This response is meaningless because I am not claiming to have a
different, "equally legitimate" interpretation of predicativity.
I am claiming that the established convention is not valid as an
analysis of *any* coherent foundational stance, and my claim is
backed up by an extensive analysis which I doubt Friedman has read.

Nik Weaver
Math Dept.
Washington University
St. Louis, MO 63130 USA
nweaver at math.wustl.edu
http://www.math.wustl.edu/~nweaver/