Powerset:
The powerset of a set is the set of all
subsets of the set. In
something close to the standard notation of set theory: Powerset(X) = {X_{i} : X_{i} C
X} = the set of sets X_{i} that are included in X. It is a convenient
way of referring to all possibilities inherent in the set. In
standard set theory this requires an axiom, namely the axiom of
powersets, which asserts that every set has a powerset 
equivalently: If one has a set, one also has the set of its
subsets.
This is easily illustrated for small sets. Thus, the set of {a, b, c}
has the powerset {Ø, {a}, {b}, {c}, {a,b}, {a,c}, {b,c}, {a,b,c}}, where
Ø represents the empty set, that may also be written as {}.
It should be noted that the elements of a powerset of a set of
elements (of any kind) are sets, and that it is not difficult
to prove that for finite sets, the number of sets in the powerset of a
set of n elements is 2^{n} i.e. 2 to the power of n. In the
above example, n=3 and 2^{3}=8. This
explains the name of the powerset and implies that the number of sets in the
powerset of a set is always greater than the number of elements in the
set.
Cantor has an elegant argument by which one can prove that if one
admits infinite sets, then for these sets also 2^{n} > n. And
this in turn implies that then there are infinities upon infinities of
ever larger infinite sets, simply by taking powersets of powersets
without limit.
