[FOM] Is Godel's Theorem surprising?

Charles Silver silver_1 at mindspring.com
Thu Dec 7 08:12:42 EST 2006

	Why should it be so surprising that PA is incomplete, and even (in a  
sense) incompletable?
Or put the other way, why should we have thought PA (or, for Godel,  
the system of Principia Mathematica and related systems) would have  
to be complete?    It has been alleged, for example, that at the time  
of Godel's proof John von Neumann had been working on proving  
*completeness* for PM or some related system?   Why would von Neumann  
have thought *intuitively* that the system could be proved complete?
	I'm not intending the above to be questions of mathematical fact.   
I'm just wondering what accounts for the shock so to speak of Godel's  
Theorem.   One answer I've read is to the effect  that everyone at  
the time thought PM was complete.  But for me, that's not  
satisfactory.  I'd like to know *why* they should have thought it was  
complete.  Did they have *intuitions* for thinking it had to be  
	I'm also wondering, though this is a separate point, whether today  
the theorem is not only not surprising, but perhaps even intuitively  
obvious.  One unsatisfactory answer would be that incompleteness is  
now not surprising because we now know it holds.  But do we now have  
distinctly different *intuitions*, aside from the proof itself  
(though of course the proof can't entirely be discounted), that  
establish, let's say the "obviousness" of the result?

Charlie Silver

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