What is Goedel's basic idea?
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From: John Jones on 27 Mar 2010 10:27 Goedel's incompleteness theorems are particular cases of a more general theorem. This theorem (Theorem A) is THEOREM A "signs that appear the same or not the same can be employed or identified in the same way." This, in turn, is premised on THEOREM B "a framework exists for which signs that appear the same or not the same can be employed or identified in the same way." EXAMPLES First example, 1) Let GO be the instruction to start and complete a statement. 2) Let GO be a statement of the completed task. Then, by theorem A, 3) A description of GO is not necessarily a sufficient description of GO. This proof is premised on Theorem B which allows that signs that appear the same, GO 1) and GO 2) in this case, are the same if we have a framework that encompasses both. The framework in this case is physical space. Both GO 1) and GO 2) are present in the same framework of physical space; therefore, by theorem A they can be employed or identified in the same way. Another example, 1) Let X be a dog's bone 2) Let X be a non-dog's bone Then, by theorem A and theorem B, X 1) and X 2) are equivalent when X belongs to the dog. In this case the framework provided by theorem B for the equivalence of X 1) and X 2)is belonging or possession. 3) A description of X cannot itself be a sufficient description of X. THE JOHN JONES CONCLUSION There are many applications of theorem A or the "equivalence of signs" theorem, and Godel employs one of these. Such ways include making cardinals and ordinals equivalent in the framework of a Cantorian picture, and elsewhere, in the equivalence of a sign with its own identity in the framework of reidentifiable spatiality (the sign written on the page). Theorem A and Theorem B are the theorems on which Goedel bases his incompleteness theorems. Once they are recognised, the Goedellian conclusion that systems are necessarily incomplete or inconsistent is seen to be itself premised on complete and consistent systems or frameworks.
From: Jesse F. Hughes on 27 Mar 2010 12:44 John Jones <jonescardiff(a)btinternet.com> writes: > Goedel's incompleteness theorems are particular cases of a more general > theorem. This theorem (Theorem A) is > > THEOREM A > "signs that appear the same or not the same can be employed or > identified in the same way." > This, in turn, is premised on > > THEOREM B > "a framework exists for which signs that appear the same or not the same > can be employed or identified in the same way." Yes! Yes! I see! Just write stuff like that and they'll give you a PhD and everything![1] Footnotes: [1] Lordy, I hope not. -- Jesse F. Hughes "My name is Apusta Malusta Cadeau and I fight bad guys. And I'm a knight." -- A. M. Cadeau (nee Quincy P. Hughes), age 4
From: John Jones on 27 Mar 2010 21:03
Jesse F. Hughes wrote: > John Jones <jonescardiff(a)btinternet.com> writes: > >> Goedel's incompleteness theorems are particular cases of a more general >> theorem. This theorem (Theorem A) is >> >> THEOREM A >> "signs that appear the same or not the same can be employed or >> identified in the same way." >> This, in turn, is premised on >> >> THEOREM B >> "a framework exists for which signs that appear the same or not the same >> can be employed or identified in the same way." > > Yes! Yes! I see! Good. I was getting eyesore. > > Just write stuff like that You could do it for me. I'm going to bed. > and they'll give you a PhD and everything![1] Accept it on my behalf please. I'm bushed. > > Footnotes: > [1] Lordy, I hope not. Good. With that support I won't need to keep awake hoping. |