[FOM] Feferman-Schutte: response to Taotao

Nik Weaver nweaver at math.wustl.edu
Mon Apr 17 00:52:00 EDT 2006

Xing Taotao wrote:

> The "predicativist" in Feferman's sense (an early sense of his)
> is not the "rational actor" in Weaver's sense, but rahter an
> idealized actor whose mind amounts just to the "autonomous
> progression". He tries to start from natural numbers and go as
> far as he can in a straight way  (that is, new objects are
> obtainable only through proofs using previously obtained objects).
> He just keeps going by using something as an instance of (*) at
> each step taken, but as he acually knows no limit on the way
> forward, he is not in a position to konw something about ALL
> instances of (*), not to mention the general (**), even if he
> is potentially capable of going through all instances of (*)
> to some extent.

I think this is a good description of the way a predicativist
is presumed to reason on the Feferman-Schutte analysis, but it
doesn't seem to really answer the objections I raised.

I objected first that individual instances of (*) are generally
not predicatively legitimate because there is an implicit appeal
to an impredicative comprehension axiom.  As far as I can tell, no
one has directly rebutted this point.

Taotao states that Feferman's predicativist's mind "amounts just to
the `autonomous progression'."  This could be a response to my first
objection --- we are simply writing in the ability to accept instances
of (*) as part of the definition of predicativism.  But he adds that
the predicativist operates according to the principle "new objects are
obtainable only through proofs using previously obtained objects", and
my point is that this is exactly the reason why comprehension axioms
strong enough to justify (*) are not available to a predicativist.

Taotao's argument amounts to simply an assertion that predicativists
can accept each instance of (*), followed by a more lengthy (and quite
correct) explanation of why this does not entail that they accept the
general implication (**).  This is just Friedman and Koskensilta's
point that (*) does not entail (**).

My question is: why can predicativists accept each instance of (*),
and why can they not accept (**)?  The Friedman/Koskensilta/Taotao
answer is, it seems to me, essentially that accepting every instance
of (*) does not oblige you to accept (**).  I don't think that's a
satisfactory answer.


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