X-Message-Number: 23895
From: "michaelprice" <>
References: <>
Subject: Godel, undeciability and the excluded middle
Date: Thu, 15 Apr 2004 09:47:51 +0100

Peter Merel wrote:

> Godel's incompleteness theorem provides a procedure that, for 
> any sufficiently powerful formal system, creates a statement in 
> that system which cannot be consistently proved by that system.
> It is generally concluded from Godel that all formal systems are 
> either incomplete or inconsistent. But in fact Godel's theorem 
> assumes the law of the excluded middle. That law holds that any 
> statement in a sufficiently powerful formal system is either true, or 
> otherwise its negation is true.

I interpret Godel's result as disproving the law of the excluded 
middle.  Of course I don't know if Godel saw it that way.
> In a formal system that includes the value "ambiguous", Godel's 
> theorem presents no difficulty. 


> Of course almost all set theory, and so almost 
> all mathematics, is based on the excluded middle. 

Don't all axioms have the value "ambiguous"?


> http://www.cryonet.org/cgi-bin/dsp.cgi?msg=22106 and its 
> sequelae, 
> there is no necessary reason to think physical infinities exist. 

Except for the infrared divergences of quantum electrodynamics,
which imply that the finite number of photons observed with our 
insensitive detectors are a subset of an infinite (aleph-null) number 
of soft, but real, photons actually emitted.  
But we've discussed this before.

Michael C Price

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