[FOM] Godel's Theorems

[Timothy Bays]

Harvey,

FWIW, I take it that the answers to 1 and 4 are pretty obviously "yes." 
  I would guess that anyone who says "no" does so for the reasons hinted 
at in your 2---i.e., they don't like to equate the intuitive notion 
"true" with any of the formal definitions mathematicians use to specify 
this notion.

Best -- Tim

On Tuesday, April 29, 2003, at 08:55  PM, Harvey Friedman wrote:

> I looked at
>
> On Floyd and Putnam on Wittgenstein on Godel,  
> http://www.nd.edu/~tbays/papers/
>
> and
>
> http://staff.washington.edu/dalexand/Putnam%20Readings/Notorious.pdf
>
> I have some specific questions. Let PA be the usual Peano Arithmetic.
> Consider the claim
>
> *) there is a true sentence in the language of PA which is not 
> provable in PA.
>
> 1. Conventional wisdom is that this is now a fully established
> theorem of mathematics (or ordinary mathematics as currently
> practiced by the overwhelming majority of mathematicians). Is there
> agreement on this?
>
> 2. For those who do not agree, do they believe that *) is not a
> mathematical statement capable of mathematical proof? E.g., this
> could be on the grounds that they do not accept the usual
> mathematical definition of "true sentence in the language of PA".
>
> 3. For those who believe that *) is a mathematical statement capable
> of mathematical proof, but do not agree with 1. Do you see a flawed
> step in the mathematical proof of *)? E.g., that it uses some
> questionable inductions and/or definitions by recursion.
>
> On a related but separate, matter,
>
> 4. Conventional wisdom is that the establishing of *) would be a very
> major event in philosophy of mathematics and/or foundations of
> mathematics. Do you agree with this?
>
> 5. For those who do not agree with 4, please elaborate.