The purpose of this paper is
to prove that all exact functors from the category 𝒮 of sets to itself are
naturally isomorphic to the identity if and only if there are no measurable
cardinals. The first step in the proof is to approximate arbitrary left-exact
endofunctors F of 𝒮 with endofunctors of a special sort, reduced powers, and to
characterize reduced powers in terms of category-theoretic properties. The next
step is to determine the effect, on the approximating reduced powers, of the
additional assumption that F preserves coproducts or coequalizers. It turns out
that preservation of coequalizers is an extremely strong condition implying
preservation of many infinite coproducts. From this fact, the main theorem follows
easily.
