FOM: Question grown from Friedman talks

[Martin Davis]

At 09:07 AM 2/3/01 -0500, Harvey Friedman wrote:
>Martin Davis wrote Fri, 02 Feb 2001 20:57:34:
>
> >At 12:22 PM 2/2/01 -0500, Harvey Friedman wrote:
> >>I believe that no conjectures in the existing literature within normal
> >>mathematics are independent of ZFC.
> >
> >I wonder whether this is simply a hunch, or whether Harvey has some
> >rational ground for this belief. As is well known, G\"odel did not believe
> >this.
>
>With all due respect, what is the evidence that Godel disagreed with me on
>this? We are talking about the actual real world real life hard nosed
>materially existing normal mathematical literature in our libraries.

 From G\"odel's Gibbs address:

"Now if one attacks [the] problem [of axiomatizing set theory], the result 
is quite different from what one would have expected. ... one is faced with 
an infinite series of axioms,
which can be extended further and further, without any end being visible 
... there can never be an end ... because the very formulation of the 
axioms up to a certain stage gives rise to the next axiom. It is true that 
in the mathematics of today the higher levels of this hierarchy are 
practically never used. ... it is not altogether unlikely that this 
character of present-day mathematics may have something to do with ... its 
inability to prove certain fundamental theorems, such as, for example, 
Riemann's hypothesis ..."

He goes on to suggest that a "set-theoretic number theory" still to be 
developed will go much further than analytic number theory has managed to 
accomplish.

I don't mean to suggest that G\"odel's views may not be wrong. It's just 
that I'm at a loss to understand on what basis Harvey just dismisses the 
possibility that this view may be right.

Martin



                           Martin Davis
                    Visiting Scholar UC Berkeley
                      Professor Emeritus, NYU
                          martin at eipye.com
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