[FOM] Disproving Godel's explanation of incompleteness

[Richard Heck]

A.P. Hazen wrote:

>    What counts as an EXPLANATION is one of the great open problems in the philosophy of science, and what counts as an explanation in MATHEMATICS is.... 
>
so hard as not even to count as an open problem yet? That's how I feel
about it. Jamie Tappenden's recent paper "Proof Style and
Understanding", available on his web page, starts to make some strides
towards an understanding of what the problem is, though, and how it
might be addressed.

>1) Th[e derivation of incompleteness from the undefinability of truth] gives no information about what the true unprovable will be like: Gödel gave a specific, Pi-1-1, example.  On the other hand, it generalizes: we can define, in PA, systems more general than conventional FORMAL DEDUCTIVE systems (e.g.: the "experimental logics" of Jeroslow (cf. his article in "J. of Philosophical Logic," v. 4 (1975))), and by Tarski's Theorem they aren't going to be complete either.  (As has been recognized for a long time: the "Syntax" chapter of Quine's 1940 "Mathematical Logic" points out that the set of protosyntactic (=, more or less, arithmètic) truths is not only not r.e. but not even "protosyntactically definable.")
>  
>
The last talk I saw Quine give was at a conference on truth held at
Boston University in the early 1990s. The talk was on the relation
between incompleteness and the undefinability of truth. It covered much
the same ground as the relevant chapter of _Mathematical_Logic_, but, as
I remember it, Quine's point was that undefinability is much stronger
than incompleteness and is ultimately the more fundamental result.

Richard Heck