[FOM] Disproving Godel's explanation of incompleteness

[praatika@mappi.helsinki.fi]

Roger Bishop Jones <rbj01 at rbjones.com>:

> Godel believed that:
> 
> (A).  The truth predicate of a language cannot be defined in that 
> language.
> 
> and
> 
> (B).  That (A) explains the incompleteness of arithmetic.


And where exactly Goedel says this ?


:

> It can be presented alternatively as:

> 	True (simpliciter) <=> true in every model of ZFC
 
> Note: also that for the purposes of this proof this semantics 
> does not need to be "correct" (whatever that might mean), it 
> need only be well-defined.  


So we could as well define e.g.

True (simpliciter) <=> a negation-free formula of L(ZFC) ?

Certainly we are here interested in extensionally correct truth-predicate, 
one that satisfies Convention T. 

 
> Conjecture:
> The truth predicate for ZFC under the provability semantics
> is definable in ZFC under that semantics.
> Hence (A) is false for ZFC under the provability semantics.

Depends on how exactly one definies "definable". For various alternatives, 
this is trivially true. 

In one good sense, it was Goedel who showed this, so it is strange ideed 
to suggest that he did not recognize this. 

> Observation:
> The incompleteness of ZFC, in the sense relevant to Godel's 
> theorem on the incompleteness of arithmetic, is a purely 
> syntactic property, hence even ZFC under the provability 
> semantics is incomplete in this sense (though under the 
> provability semantics ZFC is complete in the different sense 
> that it proves all true sentences).

Depends on how one formulates Goedel's theorem. RBJ only considers the 
part that demonstrates that there is an independent sentence. Often, 
however, it is states as follows: For every sufficiently rich theory T, 
there is a *true* sentence of L(T) which is unprovable in T. The argument 
is impotent with this formulation.  

 
> Conclusion: (B) is false (as well as (A))

Nothing RBJ says shows that. This is an equivocation fallacy, by 
redefining "true" as "derivable in ZFC". 

> [ a shorter disproof might be: (B) is false, because (A) is 
> false, presumably you can't explain anything with a falsehood.

Wrong again! A is correct, in any standard sense of "true".

> Though Godel was notoriously precise and painstaking in his work, 
> I have not seen any attempt by him to make (A) precise.
> Does anyone know whether he ever tried to make this claim precise 
> (or showed any awareness that it might be vague or false)?

Tarski made it precise, and Goedel referred to him. 

> For anyone interested, here is my explanation of incompleteness:
> 
> It is part of our requirements of a formal deductive system that 
> proofs in that system provide checkable tokens of truth,

- and what does "truth" mean here ?

 and 
> hence that proofhood be decidable and theoremhood 
> semi-decidable.
> Arithmetic truth is not decidable (as shown by Turing) but a 
> complete (in G's sense) deductive system for a language which 
> included arithmetic would yield a decision procedure (as noted 
> by Turing), hence there can be none.

A RBJ-style counterargument: define "truth" as "well-former formula", and 
look, truth is decidable. 


Note that I am not really arguing for (B), I just wanted to point out 
various questionable steps in RBJ's reasoning. 


-Panu



Panu Raatikainen
Ph.D., Academy Research Fellow,
Docent in Theoretical Philosophy

Department of Philosophy
P.O. Box 9
FIN-00014 University of Helsinki
Finland

E-mail: panu.raatikainen at helsinki.fi

http://www.helsinki.fi/collegium/eng/Raatikainen/raatikainen.htm