FOM: Goedel: truth and misinterpretations

Matt Insall montez at
Thu Nov 2 07:56:16 EST 2000

Torkel Franzen says:
   (2) Even if every even number greater than 2 is the sum of two
       primes, this is not necessarily provable in ZFC.

   You have a difficulty, then, with the use of Goldbach's conjecture
in a context such as (2). Can you explain the nature of this
difficulty?  It is insufficient to merely *claim* that we cannot
meaningfully say such things as "every even number greater than 2 is
the sum of two primes" except in certain restricted types of context,
such as "it has been mathematically proved that ...".

My comment:
Professors Kanovei and Sazonov have asserted that (2) has 
no meaning to them.  However, I expect that they will 
consider a similar statement to have meaning:

(T)  Even if the statement ``every even number greater than 
2 is the sum of two primes'' is consistent relative to 
ZFC, then this fact may not be provable.

Is it not the case that this is true?  For in fact, one 
such situation is observed in considering large cardinals.  
Is it not the case that the consistency (relative to ZFC) 
of the postulated existence of a transitive inaccessible 
cardinal is unprovable?  Thus, the assertion of T is merely 
an observation that our current knowledge about a particular 
statement in a specific formal system is incomplete enough 
that we may even consider the possibility that the statement 
of its dependence on the axioms of that formal system stands 
in relation to that formal system analogously to another 
statement which we know cannot be proved in that formal 
system, is it not?

Now, if they accept that statement T has meaning, I ask 
if they consider it to be possible for statement T to be 
true?  (Here I restrict the notion of truth to ``observed 
scientific fact'', for I am convinced that is where 
Professor Kanovei and Professor Sazonov are headed with 
their denial of the meaningfulness of others' reference 
to ``true but unprovable'' statements.)

Matt Insall

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