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From: kube@cogsci.berkeley.edu (Paul Kube)
Newsgroups: sci.philosophy.tech
Subject: Incompleteness (was: Re: Uncertainty in life)
Message-ID: <19216@ucbvax.BERKELEY.EDU>
Date: Mon, 1-Jun-87 21:14:41 EDT
Article-I.D.: ucbvax.19216
Posted: Mon Jun  1 21:14:41 1987
Date-Received: Wed, 3-Jun-87 04:07:02 EDT
References: <6762@mimsy.UUCP> <3977@sdcc3.ucsd.EDU> <8135@ut-sally.UUCP> <1192@epimass.EPI.COM>
Sender: usenet@ucbvax.BERKELEY.EDU
Reply-To: kube@cogsci.berkeley.edu.UUCP (Paul Kube)
Organization: University of California, Berkeley
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In article <1192@epimass.EPI.COM> jbuck@epimass.EPI.COM (Joe Buck) writes:
>In article <8135@ut-sally.UUCP> turpin@ut-sally.UUCP (Russell Turpin) writes:
>>Godel's *incompleteness* theorems simply say that your
>>hypothetical logician, if consistent, might come across
>>unprovable statements. In the first order predicate calculus, for
>>example, there are fully quantified statements that can neither
>>be proven or disproven. 
>
>Godel's first incompleteness theorem states that you WILL, not MIGHT,
>come up with unproveable statements if your axiom system is powerful
>enough.  

The upshot of Goedel's incompleteness theorem is the unprovability of
necessarily true statements, i.e., statements true in every model, for
sufficiently powerful systems.  That there are *some* statements your
system can't prove is a good thing: but this is the requirement of
consistency, not completeness.  First order predicate calculus is both
consistent and complete.
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--Paul.