[FOM] Godel's Theorems

[praatika@mappi.helsinki.fi]

Harvey Friedman <friedman at math.ohio-state.edu>:

> *) there is a true sentence in the language of PA which is not 
> provable in PA.
      :
> 2. For those who do not agree, do they believe that *) is not a
> mathematical statement capable of mathematical proof? E.g., this
> could be on the grounds that they do not accept the usual
> mathematical definition of "true sentence in the language of PA".

I do agree, but I think that those who don't are indeed troubled 
with "truth" of G. That is, 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.  

Some may take the whole notion of truth (in contradistinction to 
provability within a fixed systen) as a piece of illegitimate metaphysics.

Actually, I think that a mathematical truth definition does not necessarily 
help in establishing the truth of G, for it is possible to give adequate 
truth definitions in conservative extensions, eg. in ACA-0, for PA. Right?

Of course, doubting (ii) is to fail to understand what is really going on 
in Godel's proof, the role of the assumption of 1-consistency, the fact 
that any independent Pi-0-1 sentence is true etc.  (I tried to say 
something on this some time ago in FOM under the title "truth of G" or 
something.)
But it is  failure of understanding more understandable than the alleged 
denial that (i) is a well-established mathematical fact (as I said, I doubt 
that there are many who think so).   

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