[FOM] Is Godel's Theorem surprising?

praatika@mappi.helsinki.fi praatika at mappi.helsinki.fi
Fri Dec 8 10:04:29 EST 2006

Hilbert, for example, did believe that PA and other such fundamental
theories are complete. This may be in part a consequence of the following: 

At the time one did not yet understand well the dramatic difference between
first-order and higher-order logic. And because it was known that one could
characterize the standard model of arithmetic up to isomorphism (in SO
language), it was also expected that one could give a deductively complete
theory of that structure. (So what was really suprising, from this
perspective, was that SO logic is essentially incomplete; not that PA or PM is).

Of course, as we now know, there are many problems in such reasoning, but I
think something like this may have been behind the completeness expectation. 

(Hilbert probably expected further that SO logic is, not only complete but
even decidable) 

All the Best,  Panu

Panu Raatikainen

Ph.D., Academy Research Fellow,
Docent in Theoretical Philosophy
Department of Philosophy
University of Helsinki

E-mail: panu.raatikainen at helsinki.fi

	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  
> complete?
> 	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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