  We study the set of monomial ideals in a polynomial ring as an ordered set,
with the ordering given by reverse inclusion. We give a short proof of the fact
that every antichain of monomial ideals is finite. Then we investigate ordinal
invariants for the complexity of this ordered set. In particular, we give an
interpretation of the height function in terms of the Hilbert-Samuel
polynomial, and we compute upper and lower bounds on the maximal order type.
