abstract
##

Consequences of arithmetic for set theory

Lorenz Halbeisen and Saharon Shelah

###
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* is less than or equal to *D* or *D* is less
than or equal to *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 it is provable in ZF that for all
sets *A*, |seq(*A*)| is not equal to |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.