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From: hubey@pegasus.montclair.edu (H. M. Hubey)
Subject: Re: rereRe: The end of god
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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> <Cxu1yE.2vL@gpu.utcc.utoronto.ca> <383kau$5q2@scapa.cs.ualberta.ca>
Date: Thu, 20 Oct 1994 22:04:16 GMT
Lines: 47

kevin@sawnlk.cs.ualberta.ca (Kevin Wiebe) writes:

>.........................................................................
>"This complete sentence, when given to the machine of description <M> (write
>it out), will NOT spit it out on the TRUE output slot."
>.........................................................................

>Ok, bend your brain around it for a while, and you will see that it
>is IMPOSSIBLE for the machine to spit it out EITHER the TRUE or the FALSE
>slot.  It will choke and die or something.  In any case, without even
>running the machine or looking at the exact description of the machine, we
>know the above sentence will not be spit out the TRUE slot (as said above),
>therefore, the statement itself is TRUE!  We, therefore, can "see" the
>truth of the statement, but the machine cannot prove it.


But this is not the whole story.  

1) In logic you're only allowed two values. That means that if it will
not come out of the TRUE slot then it must come out of the FALSE
slot and that is not True either. So it's not TRUE and it's not FALSE.

2) One way to look at it is just as bivalent algebra and not logic (whatever it means for now).  We have parametric equations in algebra in which
the values can keep alternating.  This (the above) is one of them in
Binary/Boolean algebra. It keeps alternating values. It's the liar
paradox in disguise. 

3) However "logic" (in quotes) doesn't want values that alternate in
some parameter. This parameter might be the inferencing index or time
or something. "Logic" wants constant values -- true for all time.
Logic wants to be timeless but it keeps getting functions which
want to behave like square wave functions.

So how is it then that this simple problem (instead of saying something
as simple as "logic isn't capable of solving this problem" or that
"this problem doesn't have a constant-timeless solution") says that
"there are true statements that are not provable!"?

If truth is relative then if it's true, it must be true relative to
some system and hence provable in that system. But in your case,
the statement is not true or false since there are other ramifications
of it that are not captured by saying that "we can see that its' true".

--
						-- Mark---
....we must realize that the infinite in the sense of an infinite totality, 
where we still find it used in deductive methods, is an illusion. Hilbert,1925
