element in Bis in A. Like a proper subset, there are also proper supersets (or a super-
set that is not the entire set).
What does it mean if a set is countable?
If a set is countable (or denumerable), it means that it is finite. This also means that the
set’s members can be matched in a one-to-one correspondence—in which each element
in one set is matched exactly with one element in the second, and vice versa—with all
the natural numbers, or with a subset of the natural numbers. Mathematicians often say,
“Aand Bare in one-to-one correspondence,” or “Aand Bare bijective.” (For more about
one-to-one correspondence, see “Math Basics.”) In set theory, all finite sets are consid-
ered to be countable, as are all subsets of the natural numbers and integers. But sets
such as real numbers, points on a line, and complex numbers are not countable. 127
FOUNDATIONS OF MATHEMATICS
What is Zermelo’s axiom of choice?
A
lthough it sounds like something on a Greek restaurant menu, Zermelo’s
axiom of choice is actually a fundamental axiom in set theory. It states that
given any set of mutually exclusive nonempty sets, there is at least one set that
contains exactly one element in common with each of the nonempty sets.
This was actually one of David Hilbert’s problems that needed to be solved by
mathematicians of his day (for more about David Hilbert, see earlier in this
chapter, and in “History of Mathematics”). German mathematician Ernst
Friedrich Ferdinand Zermelo (1871–1953) took on the task, and in 1904 he
developed what is called the well-ordering theorem, which says every set can be
well ordered based on the axiom of choice.
This brought fame to Zermelo, but it was not accepted by all mathemati-
cians who balked at the lack of axiomatization of set theory (for more about
axiomatic set theory, see above). Although he finally did axiomatize set theory
and improve on his theorem, there were still gaps in his logic, especially since he
failed to prove the consistency in his axiomatic system. By 1923, German mathe-
matician Adolf Abraham Halevi Fraenkel (1891–1965) and Norwegian mathe-
matician Albert Thoralf Skolem (1887–1963) independently improved Zermelo’s
axiomatic system, resulting in the system now called Zermelo-Fraenkel axioms
(Skolem’s name was not included, although another theorem is named after
him). This is now the most commonly used system for axiomatic set theory.