FOM: truth and provability
Torkel Franzen
torkel at sm.luth.se
Tue Nov 7 04:03:58 EST 2000
Randall Holmes says:
>I have a feeling that Kanovei is actually disputing this much more
>dubious assertion:
>There is a sentence expressible in mathematical notation which is true
>(in some metaphysical sense) and not provable (in _any_ mathematical
>system).
>It is perfectly possible to dispute this assertion (which is not a
>consequence of Godel's theorem).
>Any comments from any of the parties to the discussion?
I had a comment on this in an earlier message, which didn't appear
on the list:
V.Kanovei says:
>The only sound ontological meaning of the Goedel theorem is
>that there is no theory (of certain kind) which is both
>complete and consistent. The misinterpretation of it claims
>that there exist (ontologically) sentences which are true
>(ontologically) but not provable (mathematically) -- this
>misthesis was expressed by several contributors to this list.
Although this is indeed a misconception which is often encountered,
I don't believe it has been expressed by any contributor to the list.
Rather, what has been said is that Godel's theorem shows that there
is, for any consistent extension T of PA (say), a true arithmetical
statement which is not provable in T. Indeed you yourself assented
to this, as you must, since it is a mathematical theorem. However,
it seems you wish to underline that we must not in this theorem
understand "true" in the sense of "ontologically true". As far as
I can see, your strictures in this regard have nothing in particular
to do with Godel's theorem, but apply to anything we say in or about
arithmetic.
---
Torkel Franzen
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