New essay on Goedel's Incompleteness Theorem
in [Logic]
|
Prev: Solutions manual to Mechanical Behavior of Materials, 3E Norman E Dowling
Next: Editor of Physical Review A, Dr Gordon W.F. Drake does WRONG subtraction of 8th Class mathematics.
From: Aatu Koskensilta on 30 Sep 2009 17:23 Marshall <marshall.spight(a)gmail.com> writes: > Rather, it's quite impossible to have a fruitful discussion of Groups' > ratings on usenet, that much I know from past experience. Clearly > Google Groups rating system rates only one star in the general > opinion. It seems your opinion of the Google Groups rating system is not as dismal as usual. Why is that? Do you think the ratings somehow accurately reflects this or that important quality in postings, or in general opinion? I can't really think why, given that a one star rating may be given by a lone idiot, as may a five star one. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon mann nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Marshall on 30 Sep 2009 17:34 On Sep 30, 1:26 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > > Look, to refresh your memory, here are the facts about G: I have some questions about the below. > 2. Every formula of arithmetic has an associated Godel code, > which is a natural number. If Phi is a formula, then let #Phi > be its Godel code. This associated code is arbitrary, is it not? Presumably there are many such encodings. If I understand correctly, all that is necessary is that we have a way, given Phi, to calculate #Phi, and given #Phi, a way to calculate Phi, yes? > 3. There is a provability predicate Pr for Peano Arithmetic with the > property that for any formula Phi, > If Phi is provable from the axioms of Peano Arithmetic > then > Pr(#Phi) is true > Conversely, if Phi is not provable from the axioms of > Peano Arithmetic, then > ~Pr(#Phi) is true. Is this supposed to be: If Phi is provable from the axioms of Peano Arithmetic then Pr(Phi) is true Conversely, if Phi is not provable from the axioms of Peano Arithmetic, then ~Pr(Phi) is true. Otherwise I don't see how the provability relation could be applied to a natural number. A natural number in the language of PA would be a term, not a formula, right? > 4. For every formula Phi of arithmetic, there is a corresponding > formula G with the property that PA proves > > G <-> Phi[#G] Other than the fact that for every Phi there is a G, where does G come from? How do we figure out what G is given Phi? Thanks, Marshall
From: Aatu Koskensilta on 30 Sep 2009 17:37 Marshall <marshall.spight(a)gmail.com> writes: > On Sep 30, 1:26�pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) > wrote: > >> 4. For every formula Phi of arithmetic, there is a corresponding >> formula G with the property that PA proves >> >> G <-> Phi[#G] > > Other than the fact that for every Phi there is a G, where does G come > from? How do we figure out what G is given Phi? Look up the proof of the diagonal lemma. (There's a subtle error, or omission, in Daryl's explanation. It's a good exercise to figure out where. Hint: provability does not guarantee truth.) -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon mann nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Jesse F. Hughes on 30 Sep 2009 17:47 Marshall <marshall.spight(a)gmail.com> writes: > On Sep 30, 3:40 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> Marshall <marshall.spi...(a)gmail.com> writes: >> > On Sep 29, 9:26 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: >> >> >> Nobody in their right mind cares about Google Groups ratings. >> >> > Only insane people care what others think of them? That doesn't >> > seem right. >> >> You have some sort of delusion that Google's rating system accurately >> reflects general opinion about your character? > > I didn't say "accurately reflects general opinion"; nonetheless it is > certainly a reflection of others' opinions. You interpreted "Nobody in their right mind cares about Google Groups ratings," as "Only insane people care what others think of them." Surely, you see how ridiculous this misinterpretation is. -- "Just because you're ... in a Ph.d program it does not mean that you're up to the challenge of being a real mathematician. Only those who have a purity of mind and dedication to the truth as the highest ideal have a chance." --James Harris, as Sir Galahad the Pure.
From: Marshall on 30 Sep 2009 17:54
On Sep 30, 2:23 pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Marshall <marshall.spi...(a)gmail.com> writes: > > Rather, it's quite impossible to have a fruitful discussion of Groups' > > ratings on usenet, that much I know from past experience. Clearly > > Google Groups rating system rates only one star in the general > > opinion. > > It seems your opinion of the Google Groups rating system is not as > dismal as usual. Why is that? Do you think the ratings somehow > accurately reflects this or that important quality in postings, or in > general opinion? I can't really think why, given that a one star rating > may be given by a lone idiot, as may a five star one. Ratings of individual posts are too data-poor to be of much use, however ratings of individuals are generally pretty accurate, especially if the number of ratings is high. In particular, a one-star rating is an extremely well correlated with crankery. The difference between three and four stars is more obscure. If I am popping in to a group on a topic I don't know very well, (like say math or logic) and I see a heated argument between two people, if one of the two has a one-star rating, it's quite helpful in interpreting the argument. As an aside, the last time I tried defending the Google rating system, it resulted in a firestorm of my posts getting rated one star. Ha ha! Marshall |