### 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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