Georg Cantor

Georg Cantor: The Man Who Made Infinity Countable

In January 1874 Cantor put a question to his friend Richard Dedekind: "Can a surface (say a square that includes the boundary) be uniquely referred to a line... so that for every point on the surface there is a corresponding point of the line and, conversely?" He assumed the answer was obviously no. Three years later he proved it was yes — that a whole square, and indeed a space of any number of dimensions, contains exactly as many points as a single line segment. He wrote to Dedekind at once: "I see it, but I don't believe it!"

A Merchant's Son in St Petersburg

Georg Ferdinand Ludwig Philipp Cantor was born on 3 March 1845 in St Petersburg. His father, Georg Waldemar Cantor, was a successful merchant; his mother, Maria Anna Böhm, was Russian and notably musical. In 1856, when Georg was eleven, his father's health forced the family to Germany, where they settled first in Wiesbaden.

He attended gymnasium in Frankfurt and the Realschule at Darmstadt, graduating in 1860 with exceptional marks in mathematics and trigonometry. His father intended him for engineering; Cantor extracted permission to read mathematics instead. He entered the Zürich Polytechnic in 1862, and after his father died in June 1863 transferred to the University of Berlin, where he studied under Weierstrass, Kummer and Leopold Kronecker, and befriended Hermann Schwarz. His dissertation, on indeterminate equations, was completed in 1867.

Halle, and Heine's Problem

After a stint teaching at a Berlin girls' school, Cantor joined the University of Halle in 1869 — the provincial institution where he would spend his entire career. His colleague Heine handed him a long-standing problem about the uniqueness of trigonometric series representations. Cantor solved it by April 1870.

That problem is the seed of everything. Tracking the exceptional points where such a series might misbehave forced him to think hard about infinite sets of points, and by 1872 he had defined the irrational numbers by means of convergent sequences of rationals and begun the correspondence with Dedekind that would sustain him for a decade.

More Than One Infinity

In 1874 Cantor published the result that broke the world open. He proved that the rational numbers can be put in one-to-one correspondence with the natural numbers — they are countable — while the real numbers cannot. There is not one infinity; there are at least two, and they are different sizes.

The method was the criterion he had invented: two collections are the same size if their members can be paired off exactly, no matter how large either is. It looked like a trick, and its consequences were immediate and outrageous. Cantor observed that it followed that "almost all" numbers are transcendental rather than algebraic — an existence proof for an entire class of numbers, arrived at without exhibiting a single one.

The Transfinite

His 1878 paper in Crelle's *Journal* introduced the term "denumerable sets." Kronecker's hostility nearly blocked its publication and Dedekind had to intervene; Cantor never submitted to Crelle again.

Between 1879 and 1884 he published six papers in *Mathematische Annalen* founding set theory. The 1883 *Grundlagen einer allgemeinen Mannigfaltigkeitslehre* presented the transfinite numbers as "an autonomous and systematic extension of the natural numbers" — not fudges or limits, but numbers you can order and do arithmetic with. He knew what he was doing: "I realise that in this undertaking I place myself in a certain opposition to views widely held concerning the mathematical infinite."

He denoted the transfinite cardinals with the Hebrew letter aleph, ℵ₀ being the smallest, and expressed the size of the real numbers as 2^ℵ₀. That raised the question that consumed the rest of his life. Is the continuum the very next infinity after the countable — is 2^ℵ₀ = ℵ₁? Against Kronecker's constructivism he issued his manifesto: "the essence of mathematics lies precisely in its freedom."

Breakdown, Bacon, and the Pope

Cantor suffered his first recorded depression in May 1884. The old story is that Kronecker's persecution drove him mad; the modern reading reverses the causation, holding that the depression magnified his mathematical worries rather than being produced by them. He alternated for years between believing he had proved the continuum hypothesis true and believing he had proved it false, each time finding the error.

In 1885 Mittag-Leffler persuaded him to withdraw a paper from *Acta Mathematica* on the grounds that it was about a hundred years premature. Cantor's joke was bitter: "Had Mittag-Leffler had his way, I should have to wait until the year 1984, which to me seemed too great a demand!" Their correspondence effectively ended, and with it the twelve-year creative surge that had built set theory.

In his depressive periods he turned to philosophy and literature, becoming fixed on the theory that Francis Bacon wrote Shakespeare's plays and publishing pamphlets on it in 1896–97. He was also intensely religious, and believed the transfinite numbers had been communicated to him by God. He corresponded with Catholic theologians about the nature of the infinite and wrote directly to Pope Leo XIII, hoping his mathematics might inform church doctrine.

The Last Decades

He founded the Deutsche Mathematiker-Vereinigung around 1890 and served as its first president. His final major papers, in 1895 and 1897, surveyed transfinite arithmetic and well-ordered sets. At the Zürich International Congress of 1897 he was publicly praised by Hurwitz and Hadamard and reconciled with Dedekind.

By then he had found the cracks himself, informing Hilbert in 1896 of paradoxes lurking in set theory; Burali-Forti published one independently in 1897. Personal losses came fast — his mother in October 1896, his brother in January 1899, and his youngest son on 16 December 1899. Depression dominated the rest of his life, and after 1900 he spent long periods at the Nervenklinik in Halle, taking leave from teaching in the winters of 1902–03, 1904–05 and 1907–08. He received the Royal Society's Copley Medal and an honorary degree from St Andrews, travelling there in 1911 for the university's 500th anniversary. He died of a heart attack in Halle on 6 January 1918.

Why Georg Is Called a Genius

Hilbert supplied the word himself, calling Cantor's work "the finest product of mathematical genius and one of the supreme achievements of purely intellectual human activity." The specific quality was a willingness to trust a definition over an intuition. Everyone before him treated infinity as a direction, not a quantity. Cantor proposed a criterion for sameness of size — exact pairing — applied it without flinching to infinite collections, and followed it wherever it led, including to conclusions he himself could not believe. "I see it, but I don't believe it" is the sentence of a man being dragged by his own logic, and going anyway.

The counter-case is worth stating. He never settled the continuum hypothesis, and repeatedly convinced himself he had — in both directions — before finding his mistakes, which reflects on his judgment as well as the problem's depth. Gödel and Cohen eventually showed the hypothesis is independent of the standard axioms, so the thing he chased for thirty years was not there to be caught. What he built was also unfinished and inconsistent: he discovered paradoxes in his own system, and it took Zermelo's axiomatisation, and later Fraenkel and von Neumann, to make set theory safe. His conviction that the transfinite numbers came from God, his lobbying of the Vatican and his Baconian pamphlets suggest a certainty that was not always well calibrated. And he never obtained the prestigious chair he wanted, spending forty years at Halle.

Legacy

Set theory became the foundation on which modern mathematics rests, from *Principia Mathematica* to the Zermelo–Fraenkel axioms that mathematicians still use. Hilbert's tribute is the one that stuck: no one shall expel us from the paradise Cantor created.

Achievements

• Sylvester Medal — 1904
• Notable work: list of things named after Georg Cantor
• Held posts at Martin Luther University Halle-Wittenberg
• Fields: set theory, mathematics and mathematical logic