The Final Irrefutable Proof the Godel was wrong about his incompleteness theorem
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From: David Libert on 16 Jun 2010 12:45 I could take a piece of paper and write on it 1 + 1 = 3 . That would be wrong. So maybe in 1931 when Godel wrote his incompleteness theorem, he was wrong too. -- David Libert ah170(a)FreeNet.Carleton.CA
From: R. Srinivasan on 18 Jun 2010 12:24 On Jun 16, 9:45 pm, ah...(a)FreeNet.Carleton.CA (David Libert) wrote: > I could take a piece of paper and write on it 1 + 1 = 3 .. > > That would be wrong. > > So maybe in 1931 when Godel wrote his incompleteness theorem, he was wrong too. > > -- > David Libert ah...(a)FreeNet.Carleton.CA > > Godel's theorems imply that truth is different from provability. To prove Godel wrong, we need convincing arguments that truth *is* provability. A logic which formulates such a definition of truth, as firmly tied to provability, is NAFL (non-Aristotelian finitary logic). In NAFL, Godel's theorems *cannot* possibly hold. NAFL defines the correct boundaries of finitary reasoning. Godel's reasoning falls on the infinitary side from NAFL's standpoint. The simplest way to see this is via the fact that a logical consequence of Godel's theorems is that nonstandard models of Peano Arithmetic (PA) must exist. But in NAFL, nonstandard models of PA are ruled out. Because NAFL's truth definition rules out the existence of undecidable propositions in the (NAFL version of) PA. Therefore Godel's theorems cannot hold in NAFL, essentially because they do not meet the NAFL yardstick for finitary reasoning. It is not difficult for me to see that nonstandard models are infinitary objects and that NAFL makes great sense in branding them as such. The principles of NAFL rule out classical infinitary reasoning, such, as Godel's theorems or Cantor's theory of infinite sets or Einstein's relativity theory. But remarkably, exactly these same principles end up providing strong support for the basic principles of quantum mechanics, which include quantum superposition. In fact NAFL provides a new interpretation of the wave function that refutes the recent attack on the foundations of quantum mechanics by Shahriar Afshar (whose paper was published in the prestigious journal Foundations of Physics). The NAFL interpretation and the refutation of Afshar's argument is published in the following paper of mine, which just appeared in the International Journal of Quantum Information (a mainstream and highly respected physics journal, with eminent physicists in the Editorial Board): http://dx.doi.org/10.1142/S021974991000640X This paper focuses on the logical arguments that are needed to refute Afshar. I would urge logicians and physicists to go through the first couple of sections of this paper (in particular, Section 2 "Summary of the main argument and conclusions") to understand the logical issues involved and why NAFL is eminently suitable for providing a correct logical interpretation of the "weird" phenomena of quantum mechanics. The paper is available at Cornell's Arxiv: http://arxiv.org/abs/quant-ph/0504115 There was a recent FOM discussion on falsification of Platonism, starting with this post: http://www.cs.nyu.edu/pipermail/fom/2010-April/014586.html The claim made in this post is: "If a contradiction is derived from PA, that will falsify Platonism." This is backwards. The correct statement is: "To refute PA, we must first falsify Platonism with a truth definition that firmly relates truth to provability". NAFL does exactly that. Truth *is* provability in NAFL. There is no truth for formal propositions outside of the ability to prove these propositions. That is the final, ultimate rejection of Platonism. Such a truth definition will not permit any PA-undecidable propositions, which are predicted by Godel's theorems. Hence PA is an inconsistent theory by NAFL's yardstick. In the NAFL version of PA (call this NPA) there do not exist any undecidable propositions. Regards
From: Frederick Williams on 18 Jun 2010 13:34 "R. Srinivasan" wrote: > > On Jun 16, 9:45 pm, ah...(a)FreeNet.Carleton.CA (David Libert) wrote: > > I could take a piece of paper and write on it 1 + 1 = 3 . > > > > That would be wrong. > > > > So maybe in 1931 when Godel wrote his incompleteness theorem, he was wrong too. > > > Godel's theorems imply that truth is different from provability. To > prove Godel wrong, we need convincing arguments that truth *is* > provability. A logic which formulates such a definition of truth, as > firmly tied to provability, is NAFL (non-Aristotelian finitary logic). So what? In propositional calculus the true and the provable coincide. You weren't, perchance, taking David Libert's post seriously, were you? -- I can't go on, I'll go on.
From: R. Srinivasan on 18 Jun 2010 14:22 On Jun 18, 10:34 pm, Frederick Williams <frederick.willia...(a)tesco.net> wrote: > "R. Srinivasan" wrote: > > > On Jun 16, 9:45 pm, ah...(a)FreeNet.Carleton.CA (David Libert) wrote: > > > I could take a piece of paper and write on it 1 + 1 = 3 . > > > > That would be wrong. > > > > So maybe in 1931 when Godel wrote his incompleteness theorem, he was wrong too. > > > Godel's theorems imply that truth is different from provability. To > > prove Godel wrong, we need convincing arguments that truth *is* > > provability. A logic which formulates such a definition of truth, as > > firmly tied to provability, is NAFL (non-Aristotelian finitary logic). > > So what? In propositional calculus the true and the provable coincide. > > You weren't, perchance, taking David Libert's post seriously, were you? > > -- > I can't go on, I'll go on. > > Propositional calculus is too weak to express Arithmetic. What I meant was that first-order logic should be restricted in such a way that truth coincides with provability, while still being strong enough to express Arithmetic. In such a logic (NAFL) Godel's theorems cannot hold. RS
From: Frederick Williams on 18 Jun 2010 15:55
"R. Srinivasan" wrote: > > [...] What I meant > was that first-order logic should be restricted in such a way that > truth coincides with provability, while still being strong enough to > express Arithmetic. Such systems may be found in Mathematical Logic with Special Reference to the Natural Numbers by S. W. P. Steen. -- I can't go on, I'll go on. |