[FOM] Godel, Wittgenstein etc.
praatika@mappi.helsinki.fi
praatika at mappi.helsinki.fi
Tue May 6 05:02:04 EDT 2003
Several postings here seem to suggest the the *truth* of G for PA depends
on its provability in PA^2 or ZFC or whatsoever. I think that this is
unnecessary and misleading. I've been trying to remind people that its
truth is much more elementary issue, for it is entailed by the assumption
that PA is consistent. The issue becomes important when one considers e.g.
ZFC (or whatever is the strongest system one can take seriously; perhaps
ZFC + the existence of some extremely large cardinals, or whatever).
Clearly one cannot then appeal to the provability in ZFC in order to argue
that G for ZFC is true. But we have exactly as much warrant to believe in
the truth of G(ZCF) as we have warrant to believe in the consistency of
ZFC; and similarly for other theories... At some point, this is not a
matter of mathematical proof, but a matter of the degree of confidence we
have on the consistency of a given theory (and ti may vary form theory to
theory).
Also, I think that the appeal to truth definitions is both unnecessary and
misleading. As I noted before, it is often possible to give an adequate
truth definition in a conservative extension, e.g. in NBG for ZFC (and in
ACA_0 for PA); but such an extension does *not* prove the truth of Godel
sentence. Also, the real point seems to be that (for a given system F):
Cons(F) => (-Prov([G]) & G).
It adds little to say that:
Cons(F) => (-Prov([G]) & True([G]).
A minimal theory of truth (i.e. just adding T-sentences) suffices for that
move; but it is a conservative extension too. Arguably the notion of truth
does no real work here.
Best
Panu
Panu Raatikainen
PhD., Docent in Theoretical Philosophy
Fellow, Helsinki Collegium for Advanced Studies
University of Helsinki
Address:
Helsinki Collegium for Advanced Studies
P.O. Box 4
FIN-00014 University of Helsinki
Finland
E-mail: panu.raatikainen at helsinki.fi
http://www.helsinki.fi/collegium/eng/Raatikainen/raatikainen.htm
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