### Relations between some cardinals in the absence of the Axiom of Choice

by Halbeisen and Shelah. [HlSh:699]

Bulletin Symbolic Logic, 2001

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.

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