FOM: Cantor and Hilbert knew what they were talking about

Solomon Feferman sf at Csli.Stanford.EDU
Thu Dec 4 15:12:39 EST 1997

On Thu, 4 Dec 1997 jshipman at wrote:

> Cantor and Hilbert certainly felt that CH was meaningful, and I do NOT think
> that there was anything essentially wrong with their thinking on this matter.

Quotations from Cantor, thanks to Michael Hallett *Cantorian Set Theory
and the Limitation of Size*:
"...there are transfinite cardinal numbers and transfinite ordinal types
which, just as much as the finite numbers and forms, possess a definite
mathematical uniformity, discoverable by men.  _All these particular modes
of the transfinite have existed from eternity as ideas in the Divine
intellect." (p.21)
"...these transfinite species were at the disposal of the intention of the
Creator and his absolutely inestimable will power just as were the finite
numbers." (p.23)

Cantor also thought, at least at one time, that the Well Ordering Theorem
is a "Law of Thought".

And so on...

As for Hilbert, witness his infamous "proof" of CH by proof theory in "On
the infinite", his assertion that all mathematical problems are solvable,
and in particular his call to find a decision procedure for diophantine

Where do we draw the line as to what to accept from each of these truly
great but flawed thinkers?

Sol Feferman

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