From: Lee Corbin (firstname.lastname@example.org)
Date: Thu Jun 26 2008 - 23:04:51 MDT
>> As I said before, using Gödel and Turing and making an entirely
>> reasonable analogy between axioms and goals we can conclude that there
>> are some things a fixed goal mind can never accomplish, and we can
>> predict that we can NOT predict just what all those imposable tasks are.
> Godel and Turing are overused in analogies (the class of statements
> they deal with is a narrow one). But I don't see at all analogy
> between goals and axioms....
Oh, thank you very much! Gödel's theorem has by now, I estimate
overtaken Einstein's relativity theories in incidence of flagrant misuse.
People need to keep in mind at least these three things:
1) The Gödel Incompleteness theorem of 1931 applies *only* to
(a) theories that have a certain amount of built-in arithmetic
(b) theories that are first order, in the sense that only finitary
reasoning may be used. (For example, Gentzen proved
in 1936 using transfinite numerals that arithmetic is consistent
and it's doubtful that first order is even strong enough to
capture the famous sentence "critics admire only one
another"---you need second order logic for that and
for much more in order to formalize our ordinary thinking,
if even second order is sufficient)
2) That for all we know the following systems are both complete
and consistent: (i) The Bible (ii) Shakespeare's Understanding
Of Human Nature (iii) Ayn Rand's Axioms for Everything.
3) In 1930, Gödel proved his *Completeness* theorem, showing
that first order logic (without that axiomatized arithmetical
component!) is both sound (anything you prove really is true)
and complete (if it's true, you can prove it).
So it is just hideously wrong to throw Gödel's theorem around as
in the sentence you quoted. In his marvelous, rather challenging
though entirely elementary, and very thorough little book "Gödel's
Theorem: An Incomplete Guide to Its Use and Abuse", Torkel
Franzen even demolishes Penrose's and Chaitin's loose positions.
> Random example of a fixed goal institution: a bank (or a company)
> dedicated, with single mindness, only to maximising legal profits.
> I've never heard it said that its single goal creates any godel-type
> problems. What would they be like?
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