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Topic: This Week's Finds in Mathematical Physics (Week 236)
Replies: 29   Last Post: Aug 24, 2006 9:00 AM

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 Nik Weaver Posts: 1 Registered: 8/24/06
Re: This Week's Finds in Mathematical Physics (Week 236)
Posted: Aug 24, 2006 9:00 AM

It might help if I explain where I think the "autonomous progressions"
analysis of predicativity goes wrong.

that is clearly predicatively acceptable, see which ordinals you can
access within that system, use these new ordinals to formulate stronger
systems, and repeat.

That is, we have a hierarchy of formal systems S_a, and once we've
proven that a is an ordinal (more precisely, an ordinal notation) we
get to work in S_a. There are variations but this is the basic idea.

The issue I have raised is: what justifies passing from "a is an
ordinal notation" to "S_a is valid"? Well, the argument looks very
simple. When you look at the definitions it's clear that if S_c is
valid for all c < b then S_b is valid. So let X = {b < a: S_b is not
valid}; from what I just said this set has no smallest element, hence
by the well-ordering property (i.e., a is an ordinal notation) it must
be empty. Thus S_b is valid for all b < a, and we finally conclude
that S_a is valid.

What's wrong with this argument is that forming the set X requires a
comprehension axiom --- an axiom that lets you form "the set of x such
that P(x)" for some family of properties P --- that goes way beyond
anything that anyone thinks is predicative.

Now if you look carefully at the proof that autonomous progressions get
up to Gamma_0 you find that you don't need the full strength of "S_a is
valid". What you need is just "recursive definitions of length a are
legitimate". So what's really at stake is: can a predicativist accept
that if a is well-ordered then constructions of length a are
well-defined? We're tempted to say "of course" only because we're used
to the classically trivial equivalence between induction and recursion
which is proven using an impredicative comprehension axiom.

Here's where it really gets bad for the
predicativity-is-limited-by-Gamma_0 idea. Suppose we just agree, on
whatever grounds, that predicativists can accept the statement
"induction up to a implies recursion up to a". That's *too much*!
This would allow the predicativist to see all at once that every a <
Gamma_0 is an ordinal notation, and hence that Gamma_0 is an ordinal
notation, which he's not supposed to be able to do.

The *only way* to get the Gamma_0 idea to work is to somehow show that
(1) for each a, predicativists can accept "induction up to a implies
recursion up to a" but (2) they cannot accept "for all a, induction up
to a implies recursion up to a". I am not aware of any even remotely
plausible defenses for these two claims.

I make this argument in detail in my paper "Predicativity beyond
Gamma_0", which is available at

http://www.math.wustl.edu/~nweaver/conceptualism.html

I go to some length to show how essentially the same error appears in a
variety of different analyses of predicativism. However, I think it's
fundamentally a very simple error that is obvious once you see it, and
I am somewhat baffled, and saddened, by the obstinate hostility I've
encountered in response to my paper, on the Foundations of Mathematics
discussion list and elsewhere.

(The paper is called "Predicativity beyond Gamma_0" because in the
second part of the paper I try to show that using hierarchies of truth
predicates it's possible to get well beyond Gamma_0 in a predicatively
acceptable way.)

Nik Weaver
Math Dept.
Washington University
St. Louis, MO 63130
nweaver@math.wustl.edu