FOM: Goedel: truth and misinterpretations
torkel at sm.luth.se
Tue Nov 7 03:58:18 EST 2000
Vladimir Sazonov says:
>Sorry, I am a simple sinful and I am unable to understand
>these highest matters. I only can go step-by-step, following
>explanations, definitions, deductions, calculations, etc.
Well, I'm even simpler, and as I study and ponder various interesting
results in mathematical logic, I'm led to reflect e.g. that
(1) Even if ZFC is in fact consistent, it seems that most likely
no argument proving this to everybody's satisfaction will
ever be found.
Now, we need not here consider whether the reflection (1) is *justified*,
for I am told by you and V.Kanovei - who here, I really think, adopt
a highly sophisticated philosophical approach as opposed to my naive
way of thinking - that it doesn't even make sense. So *why* doesn't it make
sense? It seems to make good sense to me.
An answer might be: it makes no sense because the infinite realm of
derivations from ZFC is illusory. And then a natural response might
be: who cares? Illusory or not, the question remains whether it is
theoretically possible to derive a contradiction from the axioms of
ZFC. To assume that it is not - i.e. that ZFC is in fact consistent -
is to make an assumption about a matter of objective fact.
At this point the exchange can take several routes. The view that
there is no such matter of objective fact has its exponents, and as
I've said before, I think Wittgenstein is the most interesting of
these. But however the argument goes, it is a separate question
whether it is even possible to do logic, mathematics, computer science,
and so on, without making the kind of use of mathematical statements
exemplified by (1).
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