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chaiti[s83,jmc]		Notes on Chaitin's paper on Goedel's theorem

Chaitin, Gregory J. (1982): Goedel's Theorem and Information,
%2International Journal of Theoretical Physics%1, Vol. 21, No. 12, 1982.

Dear Greg:

	I found your paper interesting, but I have some bones to
pick.

	1. You should take Goedel's philophical ideas seriously,
because they were a major reason why he proved the theorem rather
than someone else.  Namely, he distinguished clearly among
truth, provability and consistency.  Postivistically inclined
philosophers and mathematicians tended to systematically mix up
these ideas.  Goedel's PhD thesis proved that provability and
validity co-incided for first order logic, his famous paper proved
that they didn't co-incide for arithmetic.  When he later showed
that the continuum hypothesis was consistent with Zermelo-Frankel
set theory, this was in spite of his personal opinion that the
continuum hypothesis was false.  His ideas about mathematical
truth include the idea that mathematics should be more like physics.
He wanted a basis  for more axioms.

	2. In the first paragraph of section 2, you refer to the
statement that asserts its own provability.  You should mention
that the main technical difficulty is constructing a statement
that is intuitively equivalent to its own unprovability.  Incidentally,
the title of his paper shows that he was aware that his theorem
applied to all systems that he knew of.

	3. Your proof of "Goedel's theorem" from unsolvability of
the halting problem points up your vagueness about which Goedel
theorem you mean.  It doesn't show that consistency is unprovable,
but it does show the existence of true but unprovable statements.

	4. Your F(N) = F(N)+1 proof is the neatest I have seen.