“real: chapter_02” — 2011/5/22 — 23:05 — page3—#3
Some Finer Aspects of Set Theory 2-3
the necessity of assuring that such a choice function exists every time
we need it. Thus the axiom of choice can be formulated as follows.
Suppose S is any non-empty set and to each s ∈ S a non-empty
set A
s
is associated. Then there is a function defined on S such that
f (s) ∈ A
s
for every s ∈ S.
There are plenty of occasions in analysis where we have to make use
of the axiom of choice to generate a choice function. We would also like
to mention that for many years, mathematicians tried to give a proof
for the “Axiom of choice” and some even tried to “disprove” it. From
an important work by Kurt G¨odel in 1938 and Paul Cohen in 1963, we
now know that it is impossible ...