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Fred

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Gödel's Theorem has been used to argue that a computer can never be as smart as a human being because the extent of its knowledge is limited by a fixed set of axioms, whereas people can discover unexpected truths ... It plays a part in modern linguistic theories, which emphasize the power of language to come up with new ways to express ideas. And it has been taken to imply that you'll never entirely understand yourself, since your mind, like any other closed system, can only be sure of what it knows about itself by relying on what it knows about itself.

He proved it impossible to establish the internal logical consistency of a very large class of deductive systems - elementary arithmetic, for example - unless one adopts principles of reasoning so complex that their internal consistency is as open to doubt as that of the systems themselves ... Second main conclusion is ... Gödel showed that Principia, or any other system within which arithmetic can be developed, is essentially incomplete. In other words, given any consistent set of arithmetical axioms, there are true mathematical statements that cannot be derived from the set... Even if the axioms of arithmetic are augmented by an indefinite number of other true ones, there will always be further mathematical truths that are not formally derivable from the augmented set.

 

 

*eek* Coo *eek* You learn something every day *thumbup* although understanding is a different matter 😳 I'm only a bird after all ☹️

 

On a brighter note, my "Word for the Day" is agelast - someone who never laughs *biggrin* Well, there are fair few of those around

FH *wink*

 

Edited by - fullharness on 28 Oct 2002 12:37:16

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Gödel rocks !

 

Basically he took the old Liar or Epimenides paradox.

 

"This statement is False"

 

and turned it into mathematic formula (using a very clever thing called Godel Numbering, which I can never quite get the hang of) which can refer to itself....which says :-

 

"This fomulae is unprovable"

 

A formula that is True, but cannot be proved within the system.

 

Hence all sufficiently complex number systems contain axioms that are true, but not provable/derivable. So are incomplete.

 

 

 

 

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Well stick around in this 'seat of learning' CN and your knowledge will increase threefold *wink*

 

p.s. I've been biting my fingers every time I see your signature.... at the risk of being told orf for being a pedant, the quote should be, IIRC " you're only supposed to blow the bloody doors off" 😬

 

FH *cool*

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Gramm-ar

the science of language

If 'twere grammer, we'd have grammetically correct; grammer schools etc. From the Greek 'gramma' - a letter

 

FH *wink*

p.s. CN, I forgot to thank you for pandering to my sensibilities *tongue*

 

Edited by - fullharness on 28 Oct 2002 21:15:59

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