abstract
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Relations between some cardinals in the absence of
the axiom of choice

Lorenz Halbeisen and Saharon Shelah

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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.