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From: torkel@sics.se (Torkel Franzen)
Newsgroups: comp.ai.philosophy
Subject: Re: Minds, machines, and Godel
Message-ID: <1991Jan20.230731.579@sics.se>
Date: 20 Jan 91 23:07:31 GMT
References: <17695.9101201730@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 20 Jan 91 17:30:46 GMT

In article <17695.9101201730@s4.sys.uea.ac.uk> jrk@information-systems.
east-anglia.ac.uk (Richard Kennaway CMP RA) writes:

 >I see no reason for supposing first-order arithmetic to be consistent, other
 >than the fact that no contradictions have yet surfaced.  This is strong
 >enough evidence that I can regard its consistency as "uncontroversial"
 >(though not necessarily uncontrovertible); I have the same amount of
 >confidence (or lack of it) in the consistency of ZF (or any of its
 >equiconsistent variants).

  Hmm..on the basis of the fact that no contradiction has yet surfaced
you regard the consistency of first order arithmetic as "uncontroversial"...
By the same token, then, since no counterexample to Fermat's last theorem
has been found, you regard it as "uncontroversially" true...And for any
conjecture that has not yet been settled in ZFC, you regard it as
"uncontroversial" that it is undecidable in ZFC...

  It is of course open to anybody to claim whatever he likes concerning
what he has or has not confidence in. However, your apparent doctrine
that any statement of the form "for all k, P(k)" that has not been 
refuted is "uncontroversially" true is merely silly from the point of view
of ordinary mathematics.

  In other words: if you claim that there is evidence that first order
arithmetic is consistent, you had better come up with something more
convincing.