[FOM] Disproving Godel's explanation of incompleteness

[A.P. Hazen]

    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 I don't know if it EXPLAINS why arithmetic is 
incomplete, but the easiest PROOF that it is (proof of a weak version 
of Gödel's First Incompleteness Theorem) is as a corollary to 
Tarski's Theorem on the indefinability of truth: the set of PROVABLE 
sentences of, say, PA is definable in the language of PA, but by 
Tarski the set of TRUE ones isn't, so the two sets can't be identical 
(and if you are convinced that the axioms of PA are true, you'll 
conclude that there is a true unprovable).

     Comments.
     1) This 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 Tarsski'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.")

     2) Kenny Easwaran points out that Tarski's Theorem has 
limitations, and cites Kripke's exmple of a 3-valued language with 
its own truth predicate.  An example in a 2-valued langauge (a 
language, based on classical first-order logic) can be found in Anil 
Gupta's article in "J. of Philosophical Logic" v. 11 (1982).

     3) Kenny Easwaran also notes that this proof is given in 
Enderton's textbook.  It is also the first of three (successively 
harder) proofs of three (successively stronger forms of) Gödel's 
Theorem in Raymond Smullyan's "Gödel's Incompleteness Theorems" 
(i.m.h.o. the most user-friendly of *rigorous* accounts of the 
Incompleteness theorems and their proofs... from a  man who has also 
written some delightful *popular* accounts!).

     4) It seems overwhelmingly likely that Gödel "saw" this proof 
first.  (One eminent logician has referred to Tarski's Theorem as the 
"Gödel-Tarski Theorem" in lectures.)  He then thought to himself "But 
if I publish THIS, the mathematicians will see the word TRUTH and 
decide I'm just a rat-bag philosopher, so how can I reformulate it to 
avoid that..."  Cf. Feferman on Gödel's "caution" ("Philosophia 
Naturalis" v. 21 (1984); repr. in Feferman's "In the Light of Logic").

--

Allen Hazen
Philosophy Department
University of Melbourne