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.