[FOM] set theory in the rest of mathematics

[Harvey Friedman]

>On Tue, 25 Mar 2003, Vladimir Sazonov wrote:
>
>
>>   After appearing and formalising, set theory became a part
>>   of mathematics, but its role for the rest of mathematics
>>   was always foundational, conceptual. Now, even if it is
>>   also considered as a branch of mathematics, trying to do
>>   something on large cardinals is still rather internal
>   > business of set theory.
>
I have no doubt that large cardinals have to be used, and can be
used, to obtain deep systematic information about very elementary
discrete mathematics.

I.e., if we want to handle all problems of certain kinds, then it is
necessary and sufficient to use large cardinals. The very idea of
trying to handle all problems of certain kinds is radical and just
beginning.

In fact, there is just one good case of this  to date - the 6561
cases discussed in Equational Boolean Relation Theory on the preprint
server

http://www.mathpreprints.com/math/Preprint/show/

under my name.

In the future, there will be stronger and stronger such
classifications, using more and more large cardinals, in more and
more diverse contexts.

Such classifications will be more and more highly valued by more and
more mathematicians in the future. In this way, large cardinals will
become a demonstrably essential and valued tool in elementary
mathematics.

At some point, this should make its way into less elementary
mathematics as well. Then professional mathematicians who dislike
elementary mathematics will come on board.

This is probably premature right now, and besides, there is much
greater understanding of elementary mathematics in the general
intellectual community of nonprofessional mathematicians today than
nonelementary mathematics.