FOM: Set theory independent of foundations?

[JOE SHIPMAN, BLOOMBERG/ SKILLMAN]

Set theory is obviously a subject of its own.  While set theory unquestionably
provides the best foundation for mathematics that is currently known, I think
the reason the classification was modified to put set theory "under" foundations
is that it is impossible to work in set theory without constantly running into
foundational issues, because independent statements are everywhere.  This seems
to me a mistaken decision, though not an intolerable one.  It would become
intolerable if any of the following developments takes place:
   1) An alternative non-set-theoretic foundation of mathematics is found that
"works" at least as well as the standard set-theoretic one.
   2) The current set-theoretic foundations are augmented by the general
acceptance of higher axioms such as very large cardinals, which will make set
theory a more "ordinary" subject where one can simply claim to have proved "X"
rather than "ZFC plus axiom A implies X".
   3) Friedman's program of showing that independent statements are everywhere
in non-set-theoretic mathematics as well succeeds.              -- Joe Shipman