Georg Cantor
Mathematician · 1845-1918 · Mathematics, stage 3: Meet the mathematicians
Proved that infinite sets come in different sizes and built an arithmetic for them.
Overview
Cantor arrived at set theory from a problem about trigonometric series and found that the infinite could be measured. In 1874 he proved that the real numbers cannot be listed off against the whole numbers, and in 1891 he gave the diagonal argument, a proof short enough to fit on one page and strong enough to generalise: for any set there is a strictly larger one. From this he developed transfinite cardinals and ordinals, an arithmetic of infinite quantities. He asked whether any size lies strictly between the whole numbers and the real numbers, the continuum hypothesis, and could not settle it; later work showed the standard axioms cannot settle it either. His ideas were resisted, particularly by Kronecker, and he suffered from recurrent depression from 1884 onward. Start with a modern presentation of the diagonal argument and follow every line of it.
Where to start reading
On a Property of the Collection of All Real Algebraic Numbers 1874
The paper containing his first proof that the real numbers are not countable, which begins set theory as a subject.
Contributions to the Founding of the Theory of Transfinite Numbers 1895-1897
His mature two-part account of cardinal and ordinal numbers and the arithmetic that governs them.
Part of
Sources
- Georg Cantor MacTutor History of Mathematics
- Georg Cantor Wikipedia
- Set theory MacTutor History of Mathematics
