[FOM] Godel's Theorems

[praatika@mappi.helsinki.fi]

Charlie Volkstorf wrote:

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

[I repeat my (i) and (ii) in the end of this message.]

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. 

It is illuminating to compare this case to the undecidable sentences of 
different sort, e.g. GCH - many rational people believe that it makes no 
sense to speak about the truth value of GCH etc. But be that as it may, my 
point is that here, unlike with Pi-0-1 sentences, there is no direct route 
from undecidability to truth.

I guess that many sceptics fail to see the difference between such cases, 
and wrongly believe that also the question of truth of G for ZFC is a 
controversial metaphysical issue. 

> 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? 
Assuming the soundness of the axioms, truth here equals provability. 


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


* * * * 

What I wrote was:

... 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.