[FOM] Friedman's confusion?
Nik Weaver
nweaver at math.wustl.edu
Tue Apr 11 19:56:37 EDT 2006
Harvey Friedman purports to have identified an error in my
critique of the Feferman-Schutte analysis of predicativity.
However, the "error" is not in fact part of my argument.
Friedman has misunderstood my argument and he rephrases it
in an inaccurate manner that contains an error which is
not present in its original form, which he then identifies
as "Weaver's error". I am referring to the message
http://www.cs.nyu.edu/pipermail/fom/2006-April/010368.html
where I sketched the argument that Friedman is responding to.
I particularly object to statements of the form "Weaver says"
followed by an assertion I did not make.
I will explain again. In order for the F-S analysis to
succeed, two conditions must be met:
(1) predicativists must be able to accept every instance of
a certain deduction rule, which I called (*);
(2) predicativists must not be able to accept the corresponding
implication, which I called (**).
Friedman does not contest this comment, which should be clear
to anyone who's familiar with the F-S analysis.
My first objection is direct: instances of (*) are generally not
predicatively acceptable because they require a comprehension
axiom that predicativists don't have. Friedman has no answer to
this, no comment whatsoever.
My second objection is hypothetical: if predicativists *were*
somehow capable of accepting every instance of (*), it is hard
to see why they wouldn't accept (**) too.
Let's consider this in slightly more detail. Here's what the
F-S analysis requires: whenever a predicativist proves that some
value of a is an ordinal notation, he immediately apprehends that
every axiom of S_a is true for that value of a. However, he is not
capable of recognizing the general principle "if a is an ordinal
notation then every axiom of S_a is true".
Now I can't prove that this is impossible, and I didn't say it
was. But it's clearly implausible, under any conception of
predicativism. If he doesn't grasp the general principle, where
does his apprehension of each instance come from? A flash of
intuition? Something else?
Conceivably the objection could be overcome by explaining why
each instance of (*) is consistent with predicativist principles
but (**) is not. However, Friedman cannot do this, so instead he
accuses me of claiming that (*) ENTAILS (**). But that is not
what I said, and merely pointing out that (**) is not logically
entailed by (*) is hardly a satisfactory refutation.
All Friedman needs to do to demolish my case --- the one I
actually made --- is explain
(1) why every instance of (*) is predicatively reasonable
(2) why (**) is not predicatively reasonable.
I'm waiting.
Nik
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