**From:** John K Clark (*johnkclark@fastmail.fm*)

**Date:** Sat Jun 28 2008 - 10:12:53 MDT

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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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