[oclug] OT: Logic (was: (no subject))

[Patrick Smith]

"Francis J. A. Pinteric" wrote:
> The basic axion of science is that the world is knowable and we are
> capable of knowing it. 

I'd have thought the axiom was that we are capable of knowing at least
some things about the world, not necessarily everything.


> Notice that I use the word axion which means that
> it is an unchallenged and unproved statement taken to be proved.

In the context of mathematical logic, which you mention indirectly
below, 'twould be better to say "taken be true".

> But any
> self contained system cannot be proved by it's own set of axions,
> assumptions and deductions (this is Godel's theorem).

More accurately, it's not useful to prove the axioms of the system
within the system.  The proof is trivial ("P, therefore P"), but hardly
convincing to anyone who doesn't already believe the axioms.

And this is very different from Godel's theorem.  The incompleteness
theorem, the one he's famous for, says (loosely) that in any logical
system that is both self-consistent and sufficiently powerful to
describe arithmetic, there are statements that can be neither proven nor
disproven.  This doesn't mean we can't know whether the statements are
true or false; just that we can't give a formal proof of truth or
falsity.

There's another theorem to the effect that a self-consistent system
cannot prove itself to be consistent.  I forget if this is also due to
Godel; it's been 20 years since I studied this stuff.
-- 
patsmith at pobox.com