[FOM] FOM: Gödel's Explanation of Incompleteness.

[John Corcoran]

Gödel’s Explanation of Incompleteness. As has been established by
previous FOM Digest postings, Gödel’s Explanation of Incompleteness
(loosely, that provability is arithmetically definable but truth is not)
has been somewhat overlooked by otherwise knowledgeable and informed
scholars. It is easy to remedy this unfortunate situation (which can
lead to a distorted conception of the history of our field). In the June
1969 SCIENTIFIC AMERICAN, Alfred Tarski published a highly readable and
highly accurate popular exposition of the incompleteness theorem from
the definability viewpoint. His article is aptly titled ‘Truth and
Proof’. This article is even more accessible today. It was first
reprinted in the 1993 R.I.G. Hughes anthology A PHILOSOPHICAL COMPANION
TO FIRST-ORDER LOGIC, Indianapolis and Cambridge MA: Hackett [still in
print]. If you are lucky enough to have access to ALFRED TARSKI:
COLLECTED PAPERS edited by Steven Givant and Ralph McKenzie, Basel:
Birkhauser 1986 [long out of print], you can find it in Volume 4
[Mathematical Reviews (1991) 91h:01104].

This article is one of the clearest and most sober expositions of the
Gödel Incompleteness Theorem ever written. It is especially valuable to
scholars interested it Tarski’s anti-positivistic philosophy: it
contains clear statements of some of the philosophical principles that
guided Tarski’s thinking throughout his career in logic. For example,
see the first few paragraphs of section 2 'The Notion of Proof'.


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