The linearity of the Mitchell order

We show from an abstract comparison principle (the Ultrapower Axiom) that the Mitchell order is linear on sufficiently strong ultrafilters: normal ultrafilters, Dodd solid ultrafilters, and assuming GCH, generalized normal ultrafilters. This gives a conditional answer to the well-known question of whether a [Formula: see text]-supercompact cardinal [Formula: see text] must carry more than one normal measure of order 0. Conditioned on a very plausible iteration hypothesis, the answer is no, since the Ultrapower Axiom holds in the canonical inner models at the finite levels of supercompactness.