From: Lee Daniel Crocker (lcrocker@calweb.com)
Date: Mon Jan 13 1997 - 17:35:04 MST
>>There is nothing that prevents us from forming other axioms
>>and other systems of proof that have different unprovables.
> That could be very dangerous, it might make your system too powerful,
> so powerful it can prove things that are not true...
> [Followed by basic rehash of high-school geometry/logic]
Of course, but that wasn't my point. I was simply trying to show
how specific, narrow, limited-application results are often misapplied
as general principles. Godel's is typical, as is Darwin's. A few
results of Chaos theory are possibly some of the few cases where a
mathematical/scientific result actually does have general
application in the real world. While Godel does not really have any
implications about the limits of human knowledge as was suggested at
the start of this thread, and which I have tried to rebut, Chaos
theory does indeed do a good job of explaining why we can't predict
the weather.
Adding arbitrary axioms to a logical system changes its ability to
reflect "truth", but "truth" isn't a feature of logical systems.
Logical systems can have the properties of "consistency" or
"completeness", but not both. That--AND ONLY THAT--is Godel's result.
"Truth" is something else altogether: that's simply how we choose to
use the logical system as a model of some physical reality. Because
such mappings of logic to reality are often useful, I would not
recommend adding axioms arbitrarily except as an exercise.
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