[FOM] Godel's Theorems

[Axiomize@aol.com]

Panu Raatikainen wrote 4/30/03:

> I think it is good to separete two issues in 
> Godel's theorem (S satisfies the usual conditions):

> (i) There is a sentence G of L(S) which is neither provable or refutable in 
S.  

> (ii) moreover, G is true.

> I think that no rational person denies that (i) is a proved mathematical 
> fact, but some may be puzzled about (ii), esp. when S is some very 
> comprehensive system such as PM or ZFC or whatsoever. For weaker systems, 
> they may find (ii)  acceptable, when interpreted as meaning G is provable 
> in a stronger system, e.g. PM of ZFC.  

I think that by any reasonable definition of truth, if (i) is true then 
either G or ~G is true and thus there is a sentence (either G or ~G) that 
satisfies both (i) and (ii).  Also, if (i) is not true, then (ii) is 
meaningless, so we can't really consider these two separate issues.  You 
could instead word (i) as "The sentence G of L(S) constructed by Godel is 
neither provable nor refutable in S.", although that would leave out the 
possibility of there being alternate constructions of G that satisfy (i), and 
consequently (ii).

Charlie Volkstorf
Cambridge, MA