New essay on Goedel's Incompleteness Theorem
in [Logic]
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From: John Jones on 23 Sep 2009 18:12 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. > > The translation of the proposition P as "P is not provable in ..." is argued untenable here: http://www.scribd.com/doc/88248/Wittgenstein-on-Godel (Putnam and Floyd) though I could, and have, put my own gloss on it.
From: Aatu Koskensilta on 23 Sep 2009 18:21 John Jones <jonescardiff(a)btinternet.com> writes: > The translation of the proposition P as "P is not provable in ..." is > argued untenable here: > > http://www.scribd.com/doc/88248/Wittgenstein-on-Godel > (Putnam and Floyd) Anything at all can be argued. In this instance, Putnam and Floyd don't argue very convincingly. > though I could, and have, put my own gloss on it. Of that I'm sure. Of course, as Kreisel recounts, once the proof was explained to you in some detail, and you no longer had to rely merely on G�del's prefatory informal outline of the proof, your confusion was all cleared up -- but perhaps, having died over fifty years ago, you no longer recall Kreisel's explanation... -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon mann nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Scott H on 23 Sep 2009 21:07 On Sep 23, 5:46 pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > As said, I can't really fathom the exact nature of your confusion -- > taken literally much of what you write is simply nonsense [...] It isn't nonsense. I've read Goedel's manuscript and the proof I've given follows his. I've also been researching Goedel's Incompleteness Theorem for twelve years, since I first learned about it as a student of advanced mathematics at age 14. > That said, there is of course no pressing reason for anyone to take any > interest in the incompleteness theorems, or this or that technical > result in logic -- and unless one is willing to put in the effort and > actually study the mathematical, conceptual and philosophical issues > involved, in quite some detail, it is pointless to offer general > reflections and musings as to the significance or interpretation of such > results. Your attitude reminds me of something Wittgenstein wrote about the Liar Paradox: that "it was a useless language game, and why should anybody be excited?" To date, I don't know why we shouldn't.
From: Aatu Koskensilta on 23 Sep 2009 21:40 Scott H <zinites_page(a)yahoo.com> writes: > On Sep 23, 5:46 pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: >> As said, I can't really fathom the exact nature of your confusion -- >> taken literally much of what you write is simply nonsense [...] > > It isn't nonsense. I've read Goedel's manuscript and the proof I've > given follows his. You haven't presented any proof in the usual mathematical sense, at least in the essay on your site. The incompleteness theorems appeared very advanced, difficult to follow, at the time they were presented. As usual with such things, they are now regarded as commonplace, and various matters that were once obscure have been clarified through decades of subsequent work in proof theory and recursion theory. G�del's original paper is not best source for learning this stuff. (And your quoting G�del's original statement of the incompleteness theorem in your essay is bafflingly pointless.) > Your attitude reminds me of something Wittgenstein wrote about the > Liar Paradox: that "it was a useless language game, and why should > anybody be excited?" To date, I don't know why we shouldn't. Anyone is of course free to be excited about anything. As to the liar, pondering it has led to many important insights, in philosophy and in logic, including Tarski's theorem on undefinability of truth, Kripke's theory of grounded truth, etc. My attitude is not of much general interest, but my suggestion is by no means that we shouldn't think about various logical conundrums, or reflect on the possible philosophical significance of this or that technical result in logic. I only suggest that if one is to contribute meaningfully to our understanding of these matters, in the sense of technical philosophy or mathematical logic, it is necessary to take into account the work that's already been done, relating one's insights and ideas to the actual intellectual interests of professional philosophers and logicians. In case of "infinite reference", for example, one would expect to see some mention of Yablo's paradox and such matters. (Yablo's paradox, like many other paradoxes, can be made to do actual mathematical work, e.g. in establishing the closure ordinal for various kinds of inductive definitions.) 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 it's very easy to get stuck on some apparently brilliant but in reality vacuous insight, thinking it the bee's knees, basking in the warmth of the feeling of having really gotten to the heart of something. The possibility that this might have happened to you is something you'd do well to consider. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon mann nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Nam Nguyen on 23 Sep 2009 22:21
Aatu Koskensilta wrote: > 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 it's very easy to get stuck on some > apparently brilliant but in reality vacuous insight... Nothing further than the truth. One shouldn't go to the extreme - either way! Suppose God were so unkind to the world of Mathematics and Godel had never been born, what would have Hilbert taught us, being the Professional "know" at the time? We always should learn from what who have gone before us learnt. That doesn't necessarily mean they would _always_ be correct, whether or not they realized that, or whether or not they realized that but wouldn't want to admit it (perhaps for fear of loosing the "knowledge"?). |