[FOM] what is predicativity?

[Nik Weaver]

I hadn't intended to write anything more for now about the
Feferman-Schutte analysis of predicativity, but looking at
Harvey Friedman's message # 010471 I realized there is one
more thing I'd like to say.

Friedman wrote:

> That still allows for a system S with infinitely many axioms,
> written down by someone who is acting at least a bit beyond
> predicatively, with a claim that the predicativist can come
> to accept any finite number of the axioms, but cannot come to
> accept all at once - e.g., because the predicativist cannot
> formulate a suitably meaningful predicate to which to apply
> induction. This appears to be the approach of Feferman/Schutte.

Such an approach is highly unlikely to succeed.  Observe that
Friedman's "claim" is really two claims:

(1) a predicativist can come to accept any finite set of axioms of S;

(2) a predicativist cannot come to accept all axioms of S at once

and that he suggests a way that (2) could be made plausible but does
not say anything about (1).

This is a severe omission because however one establishes that
a predicativist can come to accept any finite fragment of S, *it
is essential that this be done in a way that cannot be recognized
as correct by the predicativist himself*.  If he could see that
there is a justification that he would accept for every axiom of
S, then he could infer that S is globally valid.

So we have the remarkable suggestion that *we* can see that every
axiom of S has a predicative justification, but the predicativist
cannot see this.  But we are given no hint as to how this could
happen.

In the specific case of the autonomous systems, the problem
becomes one of establishing that a predicativist can accept
each instance of the deduction rule (*) but cannot accept the
corresponding implication (**) (see
http://www.cs.nyu.edu/pipermail/fom/2006-April/010368.html
for details).  However, it is a problem that arises generally
whenever one is asserting both (1) and (2).  I discuss this in
more detail in Section 1.3 of "Predicativity beyond Gamma_0".

Nik