[FOM] Is Godel's Theorem surprising?

David Auerbach auerbach at unity.ncsu.edu
Sun Dec 10 19:56:55 EST 2006

On Dec 10, at 3:53 PM, Harvey Friedman wrote:

> On 12/10/06 9:19 AM, "Charles Silver" <silver_1 at mindspring.com> wrote:
>> First, thanks very much for all the interesting  and enlightening
>> responses to my question.   A couple of comments:
>> Diagonalization is not central to Godel's (first) theorem, as shown
>> by Kripke's proof of G's theorem that was published by Putnam, which
>> does not *require* diagonalization.
>> I believe this proof also shows--please correct me if I'm wrong--
>> that a specifically *mathematical* proposition (though an unusual
>> one) cannot be proved nor can its negation.
> It would be helpful to the FOM readership for you to give us a  
> reference to
> this paper by Putnam. I have serious doubts about the claims you are
> suggesting.
> Harvey Friedman

There's a "semantic" proof due to Kripke that involves enumerating  
the formulas with x as the sole free variable, adding c(i) whose  
interpretation is the ith such formula with c(i) replacing x.  The  
diagonal lemma becomes trivial, but it is still there. (the 2nd  
theorem then needs extra work to establish that the standard numerals  
are provably identical to the corresponding constants...)  But this  
can't be Putnam's Kripke's proof?

David Auerbach
Department of Philosophy & Religion
Box 8103	
Raleigh, NC 27695-8103                                    http:// 
auerbach at unity.ncsu.edu                                   http:// 

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