The Seed Order

This paper introduces the seed order, a partial order of the class of uniform countably complete ultrafilters that generalizes the Mitchell order on normal measures. Like that order, the seed order is consistently a linear ordering even under strong large cardinal assumptions. In fact, the linearity of the seed order is a feature of all known canonical inner models for large cardinal axioms. We develop the theory of the seed order under the assumption that it is linear. Augmented by large cardinal hypotheses currently out of reach of inner model theory (namely supercompactness), this linearity assumption, which we call the Ultrapower Axiom, has surprisingly strong consequences reminiscent of the structure theory of the canonical inner models.