[FOM] Re Kanovei on Gödel's Theorem

[A.P. Hazen]

   Kanovei says that the popular understanding Gödel's incompleteness
theorem is wrong, in particular that to say
	*there exist true, but unprovable sentences of PA*
	(allegedly by Goedel's theorems)
"is plain wrong in its straightforward sense...."
   He issues a challenge to the holders of the "popular" view:

	"Those who believe  in the paradigm
	*there exist true, but unprovable sentences of PA*
	in its straightforward sense
	are welcome to kindly present such a remarkable sentence
	along with a demonstration of its desired properties."

   To which I, as one who thinks the popular view contains more than a
grain of philosophical truth, want to reply:
	"This reminds me of the mother who, trying to
	get a balky child to eat its dinner, says "Think of
	all the starving children in Ethiopia who would
	just LOVE to have a bowl of spinach as nice as that
	one" to which the  brat replies "Name one.""
Starving children tend not to be known by name to residents of developed
countries, and it is of course contradictory to suppose that one could
DEMONSTRATE that a proposition is both true and  unprovable.  (This
contradiction is, I think, known to many philosophical logicians as
"Fitch's Paradox.")  So I do  not accept the challenge.

   That said, it takes more than Gödel's actual THEOREM to argue for
	*there exist true, but unprovable sentences of PA.*
The THEOREM at best shows that there is a true sentence in the language of
PA which is not provable BY MEANS of any proof formalizable within the
formal axiomatic system PA.  Reflection on Gödel's PROOF of his theorem may
yield a bit  more: for any formal system **relevantly like** PA, there is a
true sentence of PA not provable by means formalizable in that system.  It
is, in view of the the
	"25-centuries of development of mathematics under more or less
	current standards of rigor"
that Kanovei cites, moderately plausible to say that any proof beings like
us could recognize as a proof is formalizable in SOME system **relevantly
like** PA, but this is still not enough: what if the recognizable proofs
come in an endless hierarchy of increasingly strong systems **relevantly
like** PA, with no one such system containing all of them?  (At least as a
formal possibility, this is suggested by Feferman's results on ordinal
logics.)
    Needless  to say, Gödel was aware of these complexities, and discusses
them in his "Gibbs Lecture" (= *1951 in Volume III of his "Collected
Works.").  Ironically, Gödel there sides with Kanovei, and argues that no
"mechanical" model of human mathematical intelligence can be adequate!
Many of us, however, would prefer to contrapose, and argue that SINCE some
mechanical model of human mathematical intelligence has to be correct,
*there exist true, but [[humanly]] unprovable sentences of PA.*

   (From his previous posts, my guess is that Kanovei does NOT agree with
Gödel globally, but rejects an implicit assumption about the nature of
mathematical truth shared by Gödel and holders of the popular view.)

Allen Hazen
Philosophy Department
University of Melbourne