FOM: Question grown from Friedman talks
Martin Davis
martin at eipye.com
Sat Feb 3 16:39:12 EST 2001
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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