  We prove, in ZFC, that there is an infinite strictly descending chain of
classes of theories in Keisler's order. Thus Keisler's order is infinite and
not a well order. Moreover, this chain occurs within the simple unstable
theories, considered model-theoretically tame. Keisler's order is a central
notion of the model theory of the 60s and 70s which compares first-order
theories (and implicitly ultrafilters) according to saturation of ultrapowers.
Prior to this paper, it was long thought to have finitely many classes,
linearly ordered. The model-theoretic complexity we find is witnessed by a very
natural class of theories, the $n$-free $k$-hypergraphs studied by Hrushovski.
This complexity reflects the difficulty of amalgamation and appears orthogonal
to forking.
