FOM: Goedel: truth and misinterpretations
montez at rollanet.org
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 ...".
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.)
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