Re: [sl4] Re: More silly but friendly ideas

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 isnít that it canít 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 itís true, GŲdel himself proved in his PHD that itís 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 GŲdel discovered his completeness proof it
caused little splash because for years most assumed it was probably true
and most didnít care a lot about first order logic one way or the other
because itís so weak it canít even do arithmetic.

About a year later GŲdel came out with his incompleteness proof that
showed any logical system with a finite number of symbols that was
powerful enough to do arithmetic canít 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 couldnít 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 thatís true, but Gentzenís system needs an infinite number of
symbols; so unless youíre postulating an infinite and not just
astronomically large mind Gentzen is irrelevant. For any mind you
actually expect to build GŲdelís Incompleteness theorem is very relevant
indeed.

  John K Clark
    

-- 
  John K Clark
  johnkclark@fastmail.fm
-- 
http://www.fastmail.fm - IMAP accessible web-mail


This archive was generated by hypermail 2.1.5 : Wed Jul 17 2013 - 04:01:03 MDT