[FOM] necessary or sufficient?

Gabriel Stolzenberg gstolzen at math.bu.edu
Mon Feb 20 11:50:30 EST 2006

   In his message of February 20, Antonino Drago commented:

> It is interesting that the same interpretation of the relationship
> between continuum and a theory can be applied to game theory.
> Both the lemma of the supporting hyperplane and the lemma of the
> alternative for matrices can be interpreted as the requirements
> for connecting the theory to the continuum.

   Has it really been shown that they are "the requirements," rather
than merely sufficient?  To do so, you would have to show that if
either lemma is false, then the theory cannot be "connected to the
continuum."  Can you do that?

>  ....Hermann Weyl... showed that these two lemmas require at least
>  his kind of mathematics (whereas in constructive mathematics they
>  are manifesly undecidable). (more details in my  Finite game theory
>  according to constructive, Weyl's elementary, and set-theoretical
>  mathematics, Atti Fond. Ronchi, 57 (2002) 421-436).

   So, the claim is that connecting the theory to the continuum
requires the two lemmas, which in turn require Weyl's kind of
mathematics.  But, again, are these really requirements or merely
sufficient conditions?

   Finally, as for "in constructive mathematics they are manifestly
undecidable," I find it implausible but it would be very exciting
to find that it is true.

   Gabriel Stolzenberg

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