FOM: simultaneous truth of consistent statements

Vladimir Sazonov sazonov at
Thu Feb 24 13:01:38 EST 2000

Andrian-Richard-David Mathias wrote:

> The passage from Hilbert's letter to Frege that I quoted suggests to my eye that
> Hilbert was thinking of mathematics as a complete theory which is being revealed
> piecemeal to us; we add to our axioms whenever it becomes known that a possible
> addition is consistent with the axioms that are already accepted. I find no
> suggestion in his letter of an awareness that one might have to choose between
> mutually conflicting but individually consistent additions.

My feeling from that Hilbert's letter (presented by Robert Black
<Robert.Black at> Date: Fri, 18 Feb 2000 00:10:05 +0000 )
is that he had only a general methodological principle for extending
mathematics by new (consistent) concepts. I think that the possibility
of mutually conflicting concepts/theories (say, geometries!) was
so trivial for him to even mention this. (This is not intended to say
anything on the results you mentioned in your posting! This is only
on interpreting Hilbert's opinions.)

The idea of mathematics as a (complete) unique theory could arise
probably only after realizing that some such a theory like ZFC is able
to "include" all existing mathematical theories. But Hilbert started his
letter with "arbitrarily chosen axioms" and did not mention any
complete unique theory for all mathematics. It seems he (as all other
mathematicians) was thinking at that time rather in terms of many
interrelated theories, approaches, algorithms which altogether
constituted the whole mathematics.

Vladimir Sazonov

More information about the FOM mailing list