FOM: proof-theoretic ordinals (correction and follow up)
Wilfried Buchholz
buchholz at rz.mathematik.uni-muenchen.de
Thu May 14 12:11:18 EDT 1998
On Mon, 13 Apr 1998 23:22:35 Steve Simpson wrote:
> In my posting of 13 Apr 1998 10:59:01 I wrote:
> > in at least some cases the ordinal notations were later replaced by
> > direct translations or interpretations. Wasn't this the case for
> > classical and intuitionistic ID_{<omega}? Please correct me if I'm
> > wrong.
> After consultation with Harvey Friedman, I think I was wrong about
> this. According to Harvey, we don't yet know how to reduce the
> classical ID_2, ID_3, ..., ID_{<omega} to their intuitionistic
> counterparts, except via ordinal notations.
Yes, we know how to reduce.......:
On pp.222-291 of [B,F,P,S] I have proved without using ordinal notations
that ID_\nu(classical) is conservative over ID_\nu(intuitionistic,strictly
positive) for negative arithmetic sentences. The corresponding result
for ID_{<lambda} (lambda a limit ordinal) has also been proved by Sieg
in Ch.III of [B,F,P,S].
( [B,F,P,S] Buchholz, Feferman, Pohlers, Sieg: Iterated Inductive Definitions
and Subsystems of Analysis: Recent Proof-Theoretical Studies.
Springer LNM 897 (1981) )
-- Wilfried Buchholz
Name: Wilfried Buchholz
Position: Professor of Mathematics
Institution: Universit\"at M\"unchen
Research interest: proof theory, foundations of mathematics
More information: http://www.mathematik.uni-muenchen.de/~buchholz
More information about the FOM
mailing list