Last Updated：2022/12/24
The finite axiom of choice is not an axiom, but rather a theorem that can be proved from the other axioms. In contrast, there are weak forms of the axiom of choice that are not provable. One example is the axiom of countable choice, which states that if A_0,A_1,…A_n… form a denumerable set of nonempty sets, their product is nonempty. […] The axiom of countable choice is constantly used in analysis; it is often hidden so as not to sow confusion in the minds of the students (who are inclined to accept anything desired) or of the professors (who do not like to shake the foundations of the discipline).
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The
finite
axiom
of
choice
is
not
an
axiom,
but
rather
a
theorem
that
can
be
proved
from
the
other
axioms.
In
contrast,
there
are
weak
forms
of
the
axiom
of
choice
that
are
not
provable.
One
example
is
the
axiom
of
countable
choice,
which
states
that
if
A_0,A_1,…A_n…
form
a
denumerable
set
of
nonempty
sets,
their
product
is
nonempty.
[…]
The
axiom
of
countable
choice
is
constantly
used
in
analysis;
it
is
often
hidden
so
as
not
to
sow
confusion
in
the
minds
of
the
students
(who
are
inclined
to
accept
anything
desired)
or
of
the
professors
(who
do
not
like
to
shake
the
foundations
of
the
discipline).
