  If we assume the axiom of choice, then every two cardinal numbers are
comparable. In the absence of the axiom of choice, this is no longer so. For a
few cardinalities related to an arbitrary infinite set, we will give all the
possible relationships between them, where possible means that the relationship
is consistent with the axioms of set theory. Further we investigate the
relationships between some other cardinal numbers in specific permutation
models and give some results provable without using the axiom of choice.
