FOM: understanding Con(ZFC) and Goldbach's conjecture
Stephen G Simpson
simpson at math.psu.edu
Mon Sep 7 21:06:46 EDT 1998
Robert Tragesser 29 Aug 1998 12:17:29 wrote:
> There is an important sense of "understand" where it makes much
> good sense to say that we haven't understood a (mathematical)
> proposition until we have proved or disproved=
> it (or have shown that it is absolutely "undecidable"). ...
Neil Tennant 30 Aug 1998 10:23:43 objected to this, citing Goldbach's
conjecture as a counterexample:
> GC has not been proved; nor has it been disproved; nor has it
> been shown to be absolutely undedicable. YET there is no question but that
> even a mediocre mathematician understands GC precisely---that is, grasps
> *what GC says*.
Neil is of course correct in a narrow sense: we understand Goldbach's
conjecture even though its truth value is unknown. But in a larger
sense, I agree with Tragesser. There is an important sense of
"understand" (call it the strong sense) in which "to understand" a
proposition requires fairly substantial knowledge of its subject
matter, enough to have at least a reasonable conjecture about its
truth value.
Using Neil's example of Goldbach's conjecture, we understand GC in the
strong sense if for no other reason than that we can straightforwardly
verify it for even numbers up to 100 or 1000, thus making it seem more
likely to be true.
Going back to the original example of Con(ZFC), we understand it in
the strong sense because we grasp its meaning, in terms of the
predicate calculus and axioms expressing our intuitions concerning the
universe of sets. For example, we know that Con(ZFC) is true if there
exists an inaccessible cardinal kappa, because V_kappa is then a model
of ZFC.
It seems to me that this strong sense of understanding is the one that
is relevant with respect to the original issue, whether Con(ZFC) can
be understood as a proposition of number theory or finite
combinatorics. I think it cannot be so understood in the strong
sense; it can only be understood in the strong sense in terms of
predicate calculus and set theory.
-- Steve
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