[FOM] Comments on Feferman on Hellman

Nik Weaver nweaver at math.wustl.edu
Sat Jan 7 23:56:41 EST 2006

A question for Harvey Friedman:

On what grounds are you certain of the consistency of ZFC plus
the various large cardinal axioms used in your work?  And more
generally of the truth of all number theoretic consequences of
these axioms?

You prove that various combinatorial statements are equivalent to
assertions of this type and infer that this shows the "necessary
use" of large cardinals.  But that begs the question of whether
the large cardinal axioms you use are in fact consistent with ZFC
and that all number theoretic consequences of these axioms are in
fact true.  How do you know this?

A related point: you falsely claim

"36. I proved that all of these 6561 statements are provable or
refutable using large cardinals. I prove that this cannot be done
without using these large cardinals."

This is not true because obviously a different pattern of proofs
and refutations is consistent with the non-existence of these large
cardinals.  Ironically, you commit this error in the context of
lambasting Feferman for a minor inaccuracy in his paper.

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

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