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From: pindor@gpu.utcc.utoronto.ca (Andrzej Pindor)
Subject: Re: rereRe: The end of god
Message-ID: <Cxu1yE.2vL@gpu.utcc.utoronto.ca>
Organization: UTCC Public Access
References: <36vt2m$g6m@scapa.cs.ualberta.ca> <visser.781973314@galaxy.ph.tn.tudelft.nl> <CxpMF7.E6J@twisto.eng.hou.compaq.com> <37p1h3$dbo@scapa.cs.ualberta.ca>
Date: Mon, 17 Oct 1994 19:54:13 GMT
Lines: 38

In article <37p1h3$dbo@scapa.cs.ualberta.ca>,
Kevin Wiebe <kevin@swanlake.cs.ualberta.ca> wrote:
>..........
>No matter what formal system of logic you use (except, of course,
>ones too weak to even prove the basics of arithmetic), you can
>take that logic to prove that itself is incomplete.  
>
>Example: Mathematics can prove it is incomplete, itself.  We know of
>a polynomial 'D' which is of degree eight, with integer coefficients, 
>and that has eighty variables that has no solutions; but it is impossible
>to prove this within mathematics!  Namely, D(x_1, x_2, ..., x_80) = 0
>has no solutions in the natural numbers (we can show this for sure), 
                                             ^^^^^^^^^^^^^^^^^^^^^^     
>but we can't prove it mathematically - mathematics is incomplete.
>(I will not type 'D' here, because at least one of the coefficients
>is thousands of digits long.)
>
Forgive a naive question, but how "showing this for sure" is different from
"proving within mathematics"?

>Harrington, Paris, and Kirby recently discovered a true statement "Ra"
>about natural numbers that is not provable from mathematics.  
>
Did they discover that this statement is true without using mathematics?
What did they use?
Just curious.

>There ARE true logical statements out there that logic can't prove; 
>true mathematical statements that math can't prove, etc.  Get used to it.
>
>-Kevin-

Andrzej
-- 
Andrzej Pindor                        The foolish reject what they see and 
University of Toronto                 not what they think; the wise reject
Instructional and Research Computing  what they think and not what they see.
pindor@gpu.utcc.utoronto.ca                           Huang Po
