[FOM] Godel's Theorems

[Axiomize@aol.com]

Panu Raatikainen wrote 5/2/02:

> Charlie Volkstorf wrote:

>> I think that by any reasonable definition of truth, if (i) is true
>> then either G or ~G is true.

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

> If G is the Godel sentence, or more generally, any Pi-0-1 sentence,
> this is certainly the case; i.e. in reality, (i) implies (ii). I just 
wanted
> to suggest that it is exactly this step that many sceptics fail to 
understand. 

This argument applies to all of the systems to which you referred (both weak 
and strong), so I don't think it has anything to do with which system is 
being used to define truth.

True, I referred to reality whereas you referred to perception.  As far as 
perception goes, the question is, how many people (know, follow and) believe 
(i) but don't (know, follow and) believe my simple addendum?  At this point, 
we are referring to what people in general believe and why.  I think that the 
original question was, what do each of us believe, but so be it.

Regarding the question of what people in general believe and why, we could 
consider:

1. What system is S?
2. What arguments (e.g. the above) have they heard?
3. What arguments have they considered on their own?
4. How difficult is it to follow (i), (ii), and the above argument that (i) =>
 (ii)?
5. What is their intellectual capacity?

Concerning 2-4, since the proof of (i) => (ii) is much simpler than the proof 
of (i), I doubt if many people believe (i) but don't believe (ii), unless 
they are unaware of this proof of (i) => (ii).  (Godel declared his sentence 
G to be true, rather than appealing to the argument that if any given G 
satisfying (i) is not true then ~G satisfies (i) and must be true.)

>> Also, if (i) is not true, then (ii) is 
>> meaningless, so we can't really consider these two separate issues. 

> I am afraid I fail to see your point here: If S is complete, certainly the 
> question of the truth value of a sentence of L(S) makes perfect sense?

If (i) is not true, assuming that it is meaningful, then there is no sentence 
G of L(S) that is neither provable nor refutable.  Then what does it mean in 
this context to say that G is true?  I suggested that you define G 
independent of the truth value of (i), so that (ii) can refer to G in any 
case.

Charlie Volkstorf
Cambridge, MA