The
power set of a
set A is the set 2
A (
PA is used in some
textbooks) of all its
subsets: 2
A={B:B⊆A}. For
finite sets, if |A|=
n (A has
n elements) then |2
A|=2
n. This explains the funny
symbolism we picked.
For infinite sets a special axiom is needed to guarantee its existence (as a set). A problem arises, though: Even for the smallest infinite set ℵ0 (we adopt the usual set theory convention that the cardinal number is itself a set of its cardinality) we have no full understanding what all its subsets are! The set ℵ=2ℵ0 has the cardinality (also denoted c) of the continuum (the set of real numbers).
The continuum hypothesis (CH) is that 2ℵ0=ℵ1 (the second infinite cardinal). The generalised continuum hypothesis (GCH) is that 2ℵk=ℵk+1. Paul Cohen proved that (if set theory is consistent; see is mathematics consistent?) each is independent of the Zermelo Fraenkel axioms (ZF) of set theory, even with the axiom of choice (ZFC).