New essay on Goedel's Incompleteness Theorem
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From: Ross A. Finlayson on 24 Sep 2009 23:26 On Sep 24, 4:29 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Newberry <newberr...(a)gmail.com> writes: > > On Sep 23, 6:40 pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > > >> From what you say I presume you're an autodidact when it comes to the > >> incompleteness theorems. One of the dangers in being an autodidact -- > >> and I say this as a fellow autodidact -- is that it is often very > >> difficult to assess with any accuracy whether some idea, some line of > >> thought, that springs to mind, is likely to have any significance or > >> interest, from the point of view of the professional researcher; without > >> feedback from those in the know > > > Jargon and group think will not help us to solve the outstanding > > problems of the foundations of mathematics. > > Why would anyone think jargon and group think would be of any help in > solving the outstanding problem of the foundations of mathematics? > > -- > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > "Wovon mann nicht sprechen kann, darüber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus They're exceptional rationalizers. Aatu, what does Kriesel say? Thanks, Ross F.
From: Scott H on 25 Sep 2009 21:22 On Sep 24, 2:44 pm, LauLuna <laureanol...(a)yahoo.es> wrote: > On Sep 22, 11:17 pm, Scott H <zinites_p...(a)yahoo.com> wrote: > > G = ~ Pr S [~ Pr S x] > > = ~ Pr [~ Pr S [~ Pr S x]] > > = ~ Pr [~ Pr [~ Pr S [~ Pr S x]]] > > = ~ Pr [~ Pr [~ Pr [~ Pr S [~ Pr S x]]]] > > . . . > > This notation suggests there is a free variable in '~Pr S [~Pr S x]' > that can be replaced by the Gödel number of '~Pr S x'. But it is not > so, The argument in that formula is already the Gödel number of that > formula, not a free variable. I should have typed <=> instead of =. Other than that, the derivation is correct. > This makes any chain of reference terminate. You have a formula G > that, when metatheoretically interpreted, speaks about the formula G. > Full stop. G is a statement in a model of ZFC constructed *within ZFC itself*. This model of ZFC will itself have a model of ZFC, and so on, to infinity. I have chosen to call the Goedel statement of one model G and that of the next model G'. Technically, this leads to endless reference. Now, one may ask: Has self-reference really been accomplished? When we speak of G', are we really talking about the same G? Correct me if I'm wrong, but that seems to be what you're saying. I have tried to offer an intuitive account of what it would mean to add ~G as an axiom to ZFC. To do this, I have taken the endless reference of Goedel's undecidable statement at face value and explained how a supernatural number 'x' can be treated like a proof of G' and remain inductively accessible, at least in the theory, by acting like a variable.
From: Scott H on 26 Sep 2009 07:45 On Sep 24, 8:39 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > But t is not a sentence. It is a closed term in the language of > primitive recursive arithmetic, the value of which is a code for G. In > light of this, how do we make any sense of your suggestion, that > > We must remember, however, that Gödel's theorem is founded not on > self-reference but on endless reference, and that the truth value of G > could turn out to be independent of the truth value of its statement of > reference, G'. I suppose it depends on your definition of 'founded.' To ease comprehension, I've deleted 'not on self-reference' from the essay. I still say that Goedel's theorem is founded on endless reference, as t *is* a sentence in the model G is about. By the way, there appears to be a troll deliberately rating all my posts one star. What do we have here? A jealous anti-intellectual? Looks like someone took a break from feeding on the hay of herd mentality to disgrace a quality contribution to the intelligensia. Get off my thread, you lazy one-starer, and let the real mathematicians do their job.
From: David C. Ullrich on 26 Sep 2009 07:52 On Sat, 26 Sep 2009 04:45:32 -0700 (PDT), Scott H <zinites_page(a)yahoo.com> wrote: >On Sep 24, 8:39 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: >> But t is not a sentence. It is a closed term in the language of >> primitive recursive arithmetic, the value of which is a code for G. In >> light of this, how do we make any sense of your suggestion, that >> >> We must remember, however, that G�del's theorem is founded not on >> self-reference but on endless reference, and that the truth value of G >> could turn out to be independent of the truth value of its statement of >> reference, G'. > >I suppose it depends on your definition of 'founded.' To ease >comprehension, I've deleted 'not on self-reference' from the essay. I >still say that Goedel's theorem is founded on endless reference, as t >*is* a sentence in the model G is about. > >By the way, there appears to be a troll deliberately rating all my >posts one star. What do we have here? A jealous anti-intellectual? >Looks like someone took a break from feeding on the hay of herd >mentality to disgrace a quality contribution to the intelligensia. > >Get off my thread, you lazy one-starer, and let the real >mathematicians do their job. For heaven's sake. Are you suggesting that you're a real mathematician? Aatu may be a real mathematician, but _you've_ been essentially ignoring his attempts to "do his job". David C. Ullrich "Understanding Godel isn't about following his formal proof. That would make a mockery of everything Godel was up to." (John Jones, "My talk about Godel to the post-grads." in sci.logic.)
From: Scott H on 26 Sep 2009 08:55
On Sep 26, 7:52 am, David C. Ullrich <dullr...(a)sprynet.com> wrote: > For heaven's sake. Are you suggesting that you're a real > mathematician? 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. > Aatu may be a real mathematician, but _you've_ been essentially > ignoring his attempts to "do his job". No I haven't. Look at my posts: I address his challenges and I've even deleted a comment from my essay based on his feedback. How can you say that about me? |