[FOM] Godel, Wittgenstein etc.

[praatika@mappi.helsinki.fi]

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