From: John K Clark (johnkclark@fastmail.fm)
Date: Sat Jun 28 2008 - 10:12:53 MDT
On Sat, 28 Jun 2008 "Lee Corbin" <lcorbin@rawbw.com>
said:
> OF COURSE it does not mean that everything in the
> universe that is "true" (on whatever theory of
> truth you like) can be proven in pure first order logic
The trouble with first order logic isn’t that it can’t prove EVERYTHING,
the problem is that it can do almost NOTHING. It is a toy logical system
weak as tea invented long ago by professional logicians because at the
time a grown up logical system was too complicated to think about. But
yes it’s true, Gödel himself proved in his PHD that it’s consistent and
complete, that is to say any true statement that can be expressed within
it can be derived from its axioms and rules of inference and no false
statements can be. When Gödel discovered his completeness proof it
caused little splash because for years most assumed it was probably true
and most didn’t care a lot about first order logic one way or the other
because it’s so weak it can’t even do arithmetic.
About a year later Gödel came out with his incompleteness proof that
showed any logical system with a finite number of symbols that was
powerful enough to do arithmetic can’t be consistent and complete.
Nobody was expecting that and it did make a splash.
So regarding mind your objections would be correct if Mr. Jupiter Brain
worked according to first order logic, but Babbage couldn’t even make
his Analytical Engine if he used that. You also say that Gentzen came up
with a system that could do arithmetic that was consistent and complete,
and that’s true, but Gentzen’s system needs an infinite number of
symbols; so unless you’re postulating an infinite and not just
astronomically large mind Gentzen is irrelevant. For any mind you
actually expect to build Gödel’s Incompleteness theorem is very relevant
indeed.
John K Clark
-- John K Clark johnkclark@fastmail.fm -- http://www.fastmail.fm - IMAP accessible web-mail
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