Newsgroups: comp.ai.philosophy Path: utzoo!utgpu!watserv1!watdragon!violet!cpshelley From: cpshelley@violet.uwaterloo.ca (cameron shelley) Subject: Re: Minds, machines, and Godel Message-ID: <1991Jan18.181317.25833@watdragon.waterloo.edu> Sender: daemon@watdragon.waterloo.edu (Owner of Many System Processes) Organization: University of Waterloo References: <1991Jan16.035058.7465@bronze.ucs.indiana.edu> <91Jan16.135532edt.1132@neuron.ai.toronto.edu> <1991Jan17.040803.8205@bronze.ucs.indiana.edu> <1991Jan17.104913.15692@sics.se> <1991Jan17.162141.12917@watdragon. <1991Jan17.200828.376@sics.se> Date: Fri, 18 Jan 91 18:13:17 GMT Lines: 42 In article <1991Jan17.200828.376@sics.se> torkel@sics.se (Torkel Franzen) writes: >In article <1991Jan17.162141.12917@watdragon.waterloo.edu> cpshelley@violet.uwaterloo.ca (cameron shelley) writes: > > >What Goedel showed, briefly, was that for any axiomatic system T1, there > >is a Goedel number G1 which represents a statement about T1 that is > >'true' but not 'provable' in T1. > > No, Godel did not show this. What he did show was that given a formal >system T in which a certain amount of arithmetic is representable (in a >well-defined sense), we can construct a formula G which is undecidable >in T provided T is omega-consistent. Rosser strengthened this to the >theorem that we can construct a formula R which is undecidable in T >provided T is consistent. The formulation in terms of truth presupposes >a particular interpretation of the language of T. Assuming that T is >an extension of arithmetic, with the arithmetical part of T being given >its standard interpretation, it does indeed follow that G is true but >unprovable in T - provided T is consistent. Nothing follows from Godel's >theorem concerning the possibility of proving that G is true. > Well, quite true. I was addressing the effects of Goedel's theorem and should have said so explicitly. I'm not sure if I understand your last statement correctly however -- the theorem does not constrain all possible interpretations of truth (which is what I think you mean) -- but did have the effect of refuting Hilbert's conjecture that truth can be interpreted compositionally with well-formedness (if I may be allowed to paraphrase) in Frege's sense. > We have no reason whatever for claiming that we are able, in general, >to recognize the truth of the Godel sentence of an arithmetical theory, >even in those cases when the Godel sentence is true. We have the same claim to recognizing truth in this case as we do for anything else: we made it up. On the other hand, I don't think we can claim that 'truth' has been given a perfect formalisation. I suppose the question being argued here is whether ever will be able to or not. -- Cameron Shelley | "Absurdity, n. A statement of belief cpshelley@violet.waterloo.edu| manifestly inconsistent with one's own Davis Centre Rm 2136 | opinion." Phone (519) 885-1211 x3390 | Ambrose Bierce