New essay on Goedel's Incompleteness Theorem
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From: Frederick Williams on 26 Sep 2009 10:51 Scott H wrote: > A mathematician is defined as someone who is expert *or* specialized > in mathematics. Other dictionaries define it as being skilled or > learned in mathematics. In my opinion, I am specialized, skilled, and > learned in mathematics, and therefore I call myself a mathematician. Fairy snuff. Your essay is no good though. -- Which of the seven heavens / Was responsible her smile / Wouldn't be sure but attested / That, whoever it was, a god / Worth kneeling-to for a while / Had tabernacled and rested.
From: Nam Nguyen on 26 Sep 2009 13:14 Scott H wrote: > I still say that Goedel's theorem is founded on endless reference[...] Actually you might have had it upside down: Godel's theorem (say GIT) might be _unfounded_ on endless [self] reference!
From: Nam Nguyen on 26 Sep 2009 13:38 Nam Nguyen wrote: > Scott H wrote: > >> I still say that Goedel's theorem is founded on endless reference[...] > > Actually you might have had it upside down: Godel's theorem (say GIT) > might be _unfounded_ on endless [self] reference! Here seems to be a hint. On the meta level, if we want to demonstrate T is inconsistent, we'd need only to show a 1st order _syntactical_ proof of the form: (1) F /\ ~F On the other hand, if we want to _syntactically_ show T be consistent, based on the "semantic" of (1), we'd be inclined to prove this 1st order theorem: (2) F xor ~F But unlike (1), the "semantic" of (2) would _depend_ on the semantic and proof of: (F xor ~F) xor ~(F xor ~F) which in turn would depend on those of: ((F xor ~F) xor ~(F xor ~F)) xor ~((F xor ~F) xor ~(F xor ~F)) etc .... Since Godel's theorem assumes the existences of the naturals as a model of, say, Q, Godel's work basically assumes the consistency of Q. But then we have to prove the semantic of (2) be sound for Q, which means we'd degenerate into an infinite sequence of proof as mentioned above. An impossibility! Hence Godel theorem is unfounded on endless self reference about the consistency of, say, Q. Something like that.
From: Scott H on 26 Sep 2009 16:36 On Sep 26, 10:51 am, Frederick Williams <frederick.willia...(a)tesco.net> wrote: > Your essay is no good though. Why? Because none of your friends think it is?
From: Marshall on 26 Sep 2009 17:28
On Sep 26, 10:38 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > > Since Godel's theorem assumes the existences of the naturals as a model of, > say, Q, Godel's work basically assumes the consistency of Q. That doesn't sound right to me. Does the proof even mention Robinson arithmetic? Doesn't Q date from the 1950s and aren't Godel's proofs from the 1930s? I suppose it is correct to say that the technique of encoding formulas depends on the existence of arithmetic. Does that seem like a weakness? Do you have doubts about arithmetic on the naturals? Not some axiomatization of them, I mean; the actual naturals and their operators as a model. Marshall |