Part V — Chapter 18
Cantor & Dedekind — Infinity
Taming the infinite and building the foundations of modern mathematics
18.1 Dedekind Cuts
Richard Dedekind (1831–1916) gave the first rigorous definition of real numbers through partitions of the rationals. He also introduced ideals in algebraic number theory.
18.2 Cantor's Set Theory
Georg Cantor (1845–1918) showed that infinity comes in different sizes. His diagonal argument proved the reals are uncountable. He introduced cardinal and ordinal numbers, and his continuum hypothesis was later shown independent of standard axioms by Gödel and Cohen.
18.3 Opposition and Legacy
Cantor's work provoked fierce opposition from Kronecker but was defended by Hilbert: “No one shall expel us from the paradise that Cantor has created.” Today, set theory is the standard foundation for virtually all of mathematics.