A Conversation for Logical Completeness
e to the x Started conversation Jan 15, 2001
The paradox you speak of is known as Russal's Paradox and was not what Godel used in his proof. Russal's Paradox was used to show that normal Cantorian set theory was incomplete. Godel's therom used (in the propositional calculus) the statement "This statement is not provable in system S". This was the statement that would have to have it (or it's negation) added as an axiom of a logical system. But, then the system would be inconsistant.
Therfore: No system is both consistant (no statements of the form x and not x) and complete (every true statement is in the system)
The Unmentionable Marauding Pillowcase Posted Jan 21, 2001
Yes, e to the x is right. Clean up this entry a bit, make it real nice, logic is really neat and should not be on the 5 most neglected list!
Decaf Silicon Posted Jan 24, 2001
I third the motion! Always hated that phrase...
The mechanical pencils man Posted May 20, 2001
Well ii fourth it, so there!
HenryS Posted Jun 10, 2001
Almost right...but adding the godel sentence or its negation does not make the system inconsistent. I've written a bit on this towards the end of post 59 on http://www.bbc.co.uk/h2g2/guide/F53777&thread=76574&skip=40&show=20
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