Foundations of Mathematics

Set Theory

What is Zermelo’s axiom of choice?

Although 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 common with each of the nonempty sets.

This was one of David Hilbert’s problems that needed to be solved by mathematicians of his day (for more about David Hilbert, see elsewhere in this chapter, and in “History of Mathematics”). German mathematician Ernst Friedrich 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 mathematicians 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 mathematician Adolf Abraham Halevi Fraenkel (1891–1965) and Norwegian mathematician 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.


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