Formal Sciences & Matter / Mathematics

Zermelo-Fraenkel Set Theory (ZFC)

Zermelo and Fraenkel (1908-1922) wrote a short list of axioms for sets that avoids Russell's paradox; ZFC is still the standard foundation on which most of mathematics is built.

Bertrand Russell found in 1901 that unrestricted talk about sets contradicts itself, for example the set of all sets that do not contain themselves. Ernst Zermelo proposed the first axiom system in 1908; Abraham Fraenkel and Thoralf Skolem improved it in 1922, and with the axiom of choice it is called ZFC. Because ZFC states exactly what may be assumed, Kurt Gödel (1938) and Paul Cohen (1963) could show that the continuum hypothesis, a question about sizes of infinity, can neither be disproved nor proved from it.