Notes

Section 9: Implications for Mathematics and Its Foundations

Set theory [and axioms]

Basic notions of finite set theory have been used since antiquity—though became widespread only after their introduction into elementary mathematics education in the 1960s. Detailed ideas about infinite sets emerged in the 1880s through the work of Georg Cantor, who found it useful in studying trigonometric series to define sets of transfinite numbers of points. Several paradoxes associated with infinite sets were quickly noted—a 1901 example due to Bertrand Russell being to ask whether a set containing all sets that do not contain themselves in fact contains itself. To avoid such paradoxes Ernst Zermelo in 1908 suggested formalizing set theory using the first seven axioms given in the main text. (The axiom of infinity, for example, was included to establish that an infinite set such as the integers exists.) In 1922 Abraham Fraenkel noted that Zermelo's axioms did not support certain operations that seemed appropriate in a theory of sets, leading to the addition of Thoralf Skolem's axiom of replacement, and to what is usually called Zermelo–Fraenkel set theory (ZF). (The replacement axiom formally makes the subset axiom redundant.) The axiom of choice was first explicitly formulated by Zermelo in 1904 to capture the idea that in a set all elements can be ordered, so that the process of transfinite induction is possible (see page 1160). The non-constructive character of the axiom of choice has made it always remain somewhat controversial. It has arisen in many different guises and been useful in proving theorems in many areas of mathematics, but it has seemingly peculiar consequences such as the Banach–Tarski result that a solid sphere can be divided into six pieces (each a non-measurable set) that can be reassembled into a solid sphere twice the size. (The nine axioms with the axiom of choice are usually known as ZFC.) The axiom of regularity (or axiom of foundation) formulated by John von Neumann in 1929 explicitly forbids sets which for example can be elements of themselves. But while this axiom is convenient in simplifying work in set theory it has not been found generally useful in mathematics, and is normally considered optional at best.

A few additional axioms have also arisen as potentially useful. Most notable is the Continuum Hypothesis discussed on page 1127, which was proved independent of ZFC by Paul Cohen in 1963. (See also page 1166.)

Note that by using more complicated axioms the only construct beyond predicate logic needed to formulate set theory is . As discussed on page 1176, however, one cannot avoid axiom schemas in the formulation of set theory given here. (The von Neumann–Bernays-Gödel formulation does avoid these, but at the cost of introducing additional objects more general than sets.)