New essay on Goedel's Incompleteness Theorem
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From: Scott H on 21 Sep 2009 10:33 I give a concise, formal proof and a short intuitive description of ZFC + ~G. http://www.hoge-essays.com/incompleteness.html Any constructive feedback is welcome.
From: George Greene on 21 Sep 2009 10:44 On Sep 21, 10:33 am, "Scott H" <nospam> wrote: > Any constructive feedback is welcome. The first constructive improvement you could make would be not to over-claim in the first sentence. "we know for a fact that seven circles touch, that ð is greater than 3, and that there are infinitely many prime numbers" is missing the point. We may "know" these things, but "for a fact" is just from the completely wrong realm. These things are NOT facts. They follow FROM AXIOMS in MODELS OF those axioms, and if the axioms FROM WHICH these things follow are not "facts", then these things are not facts either. It may be a "fact" that these theorems follow from the axioms, but that is not even the same KIND of claim. "Fact" is just not an appropriate kind of word to be using here. And "Certainty" is NOT EVEN RELEVANT to this whole set of questions.
From: George Greene on 21 Sep 2009 10:47 On Sep 21, 10:33 am, "Scott H" <nospam> wrote: The gross over-generalizations are gross because they are over-. "This language allows even the most complicated statements to be written concisely to the occasional benefit of the reader's comprehension." is just silly. There is no such thing as "this langauge". There are TONS AND TONS OF DIFFERENT formal languages. Unfortunately, the usual definitions of Formal languages depend in a very critical way on an INformal conception of "all finite" iterations of some tactic, and since this is an infinite number of finite things, certain concepts are sort of "sneaking" in. The systems to which Godel's theorem applies are basically self-contained. They do not have anything to do with certain knowledge about the outside world.
From: Frederick Williams on 21 Sep 2009 10:59 Scott H wrote: > > I give a concise, formal proof and a short intuitive description of ZFC + > ~G. > > http://www.hoge-essays.com/incompleteness.html > > Any constructive feedback is welcome. What have Pr and Pr(a,b) got to do with the price of fish? -- 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: |-|erc on 21 Sep 2009 11:01
"George Greene" <greeneg(a)email.unc.edu> wrote On Sep 21, 10:33 am, "Scott H" <nospam> wrote: > Any constructive feedback is welcome. The first constructive improvement you could make would be not to over-claim in the first sentence. "we know for a fact that seven circles touch, that � is greater than 3, and that there are infinitely many prime numbers" is missing the point. We may "know" these things, but "for a fact" is just from the completely wrong realm. These things are NOT facts. They follow FROM AXIOMS in MODELS OF those axioms, and if the axioms FROM WHICH these things follow are not "facts", then these things are not facts either. It may be a "fact" that these theorems follow from the axioms, but that is not even the same KIND of claim. "Fact" is just not an appropriate kind of word to be using here. And "Certainty" is NOT EVEN RELEVANT to this whole set of questions. --------------------------------------------------------- This is the same line Penrose uses, that you just have to draw the line somewhere where a fact is a fact. And people here seem to follow the axiom->fact metaphor. But I think it's overcautious. (some) Platonic knowledge should be verifiable without the brain in a vat dilemma. Herc |