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Gödel proved the incompleteness of Arithmetic.  Specifically, he demonstrated that in any axiomatic system capable of Arithmetic there are statements that can neither be proved or disproved within the axioms.

To fully understand what that meant and means requires an understanding of what Logic and Mathematics were 'on about' from around 1870 to l933.  To (greatly) simplify, Frege, Hilbert, Russell, Whitehead, & All That Crew were attempting to construct an intellectual tool that would always, when correctly used, derive a True and Valid answer.  

Gödel's Proof = You Can't.

She believed in nothing; only her skepticism kept her from being an atheist. -- Jean-Paul Sartre

by ATinNM on Wed Jan 2nd, 2008 at 10:56:09 PM EST
[ Parent ]
Yes, and it failed on "this statement is false". That's the basic point of Gödel's proof. I don't know how they expected to attribute a truth value to that statement, those formal logicians ; but that you can't attribute a truth value to it sounds even a bit reconforting... Logic doesn't get out of the realm of human experience about truth values.

Un roi sans divertissement est un homme plein de misères
by linca (antonin POINT lucas AROBASE gmail.com) on Thu Jan 3rd, 2008 at 04:13:57 AM EST
[ Parent ]
Many antinomies (paradoxes,) such as that one - a variation on wossname's 'All Cretans are liars ...' - have an oscillating first order Truth Value with the TV changing depending on the step: On True Then False Then True & etc XOR On False Then True Then False & etc.  Since they are infinitely oscillating there is no resolution possible.  

By moving to the Second Order, on the other hand, the True/False OR False/True oscillation is, sometimes, capable of resolution.

But that, very often, takes you outside the original axiomatic system.

She believed in nothing; only her skepticism kept her from being an atheist. -- Jean-Paul Sartre

by ATinNM on Thu Jan 3rd, 2008 at 05:27:32 AM EST
[ Parent ]
"Arithmetic is consistent"

That was the sailing-over-the-edge moment.  

Now it may be that arithmetic really is consistent, but that is another truth function that is "uncomputable."  

The Fates are kind.

by Gaianne on Thu Jan 3rd, 2008 at 04:48:01 PM EST
[ Parent ]


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