FOM: Goedel: truth and misinterpretations
Torkel Franzen
torkel at sm.luth.se
Thu Nov 2 02:15:24 EST 2000
V. Kanovei says, with reference to
(2) Even if every even number greater than 2 is the sum of two
primes, this is not necessarily provable in ZFC.
>Mathematically, (2) is meaningless (and basically shows that
>he who writes (2) either has no proper idea of mathematics at all
>or does not bother to present his ideas in proper form).
>Indeed, "A is not necessarily B" means, in standard mathematical
>language, that there is an example of A which does not belong to B,
>e.g. "an arbitrary group IS NOT NECESSARILY an abelian group".
(2) is indeed not a mathematical statement. The same is true of other
similar observations, e.g.
(3) Even if this number is in fact composite, there is no guarantee
that we can ever factor it.
(4) Theoretically, it's perfectly possible that P=NP is true although
unprovable in ZFC, which would make it unlikely that it can ever
be settled to everybody's general satisfaction.
(5) Even if ZFC is in fact consistent, it may well be the case that no
generally convincing consistency proof for ZFC can be given.
Now, it would appear that you regard statements like (2)-(5) as
senseless (or "fraudulent", or whatever similar term you would apply).
However, it is perfectly pointless merely to forcefully affirm this
point of view, and to to complain that (2) is not a mathematical
statement. Do you have any actual arguments to present?
Some of your comments seem to be a bit disparaging of philosophy, which
is more than a little incongruous, since your own comments are very much
in the tradition of philosophy, and in fact in the tradition of radically
critical or "heroic" philosophy. I have commented before on the traditional
weaknesses of this approach.
Similar comments apply to the added reflections of V.Sazonov. I neglected
in my earlier remarks to take into account that he also regards such statements
as "ZFC is consistent" as "ambiguous", and would therefore correct many other
kinds of statements commonly made in or about mathematics.
----
Torkel Franzen
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