FOM: The meaning of truth
Kanovei
kanovei at wmwap1.math.uni-wuppertal.de
Mon Nov 6 09:37:39 EST 2000
> From: JoeShipman at aol.com
> Date: Sun, 5 Nov 2000 15:50:28 EST
> 1) If you accept proofs from ZFC as valid, then why is the standard
> definition of truth for arithmetical sentences, as commonly formalized in
> ZFC, insufficient?
Insufficient to yield a sentence (ontologically) true but
mathematically unprovable.
Let us look how your
"the standard definition of truth for arithmetical sentences,
as commonly formalized in ZFC"
works in this case.
Take any arithmetical sentence, say A, known to be
unprovable in ZFC. Applying "the standard definition", we
shall be left with another formula, say Sat_N(A), which, in
correct manner, describes that A "is true" in N.
Both A and Sat_N(A) are ZFC-formulas (A even a Peano formula),
whose equivalence is easily provable in ZFC, hence, the new
formula is as well dead unprovable as the original A.
Hard remainder: instead of one unprovable formula we have now
two of them, mathematically equivalent and both unprovable,
as clueless regarding the truth of A as before.
> I am willing to grant that the notion of "truth" I
> attribute to such sentences is inferior to the kind of "truth" you are able
> to attribute to sentences that have been mathematically proven.
If you are going to have in mind, while saying "true",
in reality notions like
*almost surely true*,
*nobody can imagine this will ever be found false*
*999 of all 1000 necessary lemmas have already been proved*
*a random sample of all found mistakes has been removed*
*is supported by all physical evidence available nowadays*
(I intentionally formulate the idea in different ways)
and the like
as an admissible (albeit "inferior") meaning of *true*
then the main problem disappears, as I readily agree that
there are sentences mathematically unprovable but such that
*not more than a handful of specialists in good mood will
expect them to ever fail*, Con ZFC the first candidate.
But neither Goedel's theorem nor its misinterpretation in
question contain any reservations that could be implicated
as that "true" may mean something other than *just true*.
V.Kanovei
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