[FOM] Disproving Godel's explanation of incompleteness
a.hazen at philosophy.unimelb.edu.au
Thu Oct 20 01:59:49 EDT 2005
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).
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").
University of Melbourne
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