Consequences of arithmetic for Set theory

by Halbeisen and Shelah. [HlSh:488]
J Symbolic Logic, 1994
In this paper, we consider certain cardinals in ZF (set theory without AC, the Axiom of Choice). In ZFC (set theory with AC), given any cardinals {C} and {D}, either {C} <= {D} or {D} <= {C}. However, in ZF this is no longer so. For a given infinite set A consider Seq (A), the set of all sequences of A without repetition. We compare |Seq (A)|, the cardinality of this set, to |{{P}}(A)|, the cardinality of the power set of A . What is provable about these two cardinals in ZF? The main result of this paper is that ZF |- for all A: |Seq(A)| not= |{{P}}(A)| and we show that this is the best possible result. Furthermore, it is provable in ZF that if B is an infinite set, then |fin(B)|<|{{P}}(B)|, even though the existence for some infinite set B^* of a function f from fin(B^*) onto {{P}}(B^*) is consistent with ZF.

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