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From: torkel@sics.se (Torkel Franzen)
Newsgroups: comp.ai.philosophy
Subject: Re: Minds, machines, and Godel
Message-ID: <1991Jan18.175633.3319@sics.se>
Date: 18 Jan 91 17:56:33 GMT
References: <15303.9101181150@s4.sys.uea.ac.uk>
Sender: news@sics.se
Organization: Swedish Institute of Computer Science, Kista
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In-Reply-To: jrk@information-systems.east-anglia.ac.uk's message of 18 Jan 91 11:50:22 GMT

In article <15303.9101181150@s4.sys.uea.ac.uk> jrk@information-systems.
east-anglia.ac.uk (Richard Kennaway CMP RA) writes:

   >Eh?  How?

  We mustn't take this talk of seeing and realizing too seriously. The
system of Principia Mathematica, if we mean what Russell and Whitehead
actually wrote, is tricky even to formalize in a way that makes the
assertion that it is consistent into a mathematical statement. (It can
be done in a reasonable way, but few people have any occasion or
reason to delve into the details, e.g. regarding the axiom of
reducibility.) In the case of an unproblematically formal system such
as ZFC, its consistency is not known to be provable in any
mathematical sense, and whether or not one regards it as evidently
consistent is very largely a matter of personal inclination and
opinion. A serious examination of the concept 'humanly provable' will
show that there is no well-defined totality of arithmetical statements
that are 'humanly provable' in a wide sense, or the truth of which we
can 'realize' or 'see'.

  In the present context, however, we are pretending that there is
such a well-defined concept of 'humanly provable arithmetical
statement' or 'humanly realizable arithmetical truth', and we're taking
a very liberal view of what is thus 'realizable'.  I think it's of
interest to note that even on this assumption, there is nothing in
Godel's theorem that implies that the 'humanly provable' statements
are not recursively enumerable.