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From: chalmers@bronze.ucs.indiana.edu (David Chalmers)
Newsgroups: comp.ai.philosophy
Subject: Re: Minds, machines, and Godel
Message-ID: <1991Jan18.184830.5723@bronze.ucs.indiana.edu>
Date: 18 Jan 91 18:48:30 GMT
References: <15303.9101181150@s4.sys.uea.ac.uk>
Organization: Indiana University, Bloomington
Lines: 26

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

>[I wrote]
>>OK, now we're homing in on something closer to the point.  Judgments of the
>>truth of Godel sentences are parasitical on judgments of the consistency
>>of the given system.  And judgments of consistency may be hard.  We can judge
>>that Principia Mathematica is consistent fairly straightforwardly
>
>Eh?  How?

Surely this isn't too controversial.  To make it even less so, make it just
a simple modern-style first-order system that axiomatizes our arithmetic
intuitions, by formalizing addition, multiplication, induction etc in the
normal ways.  Presumably the statement that this system is consistent is as
well-founded as most of our beliefs about arithmetic.  If it wasn't, the
whole of mathematics would be in danger of collapse.

Principia Mathematica is similar, though a little more baroque.  Note that
I'm only considering that part of PM that is relevant to generating
statements of first-order arithmetic.  I think it's straightforward to see
the equivalence of this system to other more stripped-down systems.

-- 
Dave Chalmers                            (dave@cogsci.indiana.edu)      
Center for Research on Concepts and Cognition, Indiana University.
"It is not the least charm of a theory that it is refutable."