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> <JMC.91Jan16213907@DEC-Lite.Stanford.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
