RELATIONS BETWEEN SOME CARDINALS IN THE ABSENCE OF THE AXIOM OF CHOICE

Abstract. 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. §1. Introduction. Using the axiom of choice, Felix Hausdorff proved in 1914 that there exists a partition of the sphere into four parts, S = A ˙ ∪ B ˙ ∪ C ˙ ∪ E, such that E has Lebesgue measure 0, the sets A, B, C are pairwise congruent and A is congruent to B ˙ ∪ C (cf. [9] or [10]). This theorem later became known as Hausdorff’s paradox. If we want to avoi