Path: utzoo!utgpu!news-server.csri.toronto.edu!cs.utexas.edu!sun-barr!apple!snorkelwacker.mit.edu!bloom-beacon!eru!hagbard!sunic!sics.se!sics.se!torkel From: torkel@sics.se (Torkel Franzen) Newsgroups: comp.ai.philosophy Subject: Re: Minds, machines, and Godel Message-ID: <1991Jan17.200828.376@sics.se> Date: 17 Jan 91 20:08:28 GMT 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. Sender: news@sics.se Organization: Swedish Institute of Computer Science, Kista Lines: 28 In-Reply-To: cpshelley@violet.uwaterloo.ca's message of 17 Jan 91 16:21:41 GMT 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. >I think Penrose's arguement is that it is not. He might say: "A human >is able to perceive the truth (a semanitc property) of the various >structures Gx ("syntactic" entities) indicated by the incompleteness >theorem, so that for the human syntactic validity is not isomorphic >with semantic validity. 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.