Georg Cantor proved some infinities are bigger, and paid a heavy price
Georg Cantor showed there are more real numbers than counting numbers, which means infinity comes in different sizes, endlessly. Leopold Kronecker called him a scientific charlatan and a corrupter of youth. Decades later David Hilbert vowed that no one would expel mathematicians from the paradise Cantor had built.
Cantor was born in Saint Petersburg on 3 March 1845, the eldest of six children, and was considered an outstanding violinist. His Danish father, a stock-exchange broker, fell ill in 1856, and the family moved to Germany for a milder climate, settling first in Wiesbaden and then Frankfurt. After a dissertation on number theory at Berlin in 1867, Cantor joined the University of Halle, where he remained his entire career and became a full professor at 34.
His great insight was to compare sets by pairing their members one to one. Using it he defined infinite and well-ordered sets, built the arithmetic of cardinal and ordinal numbers, and proved that algebraic numbers can be counted while transcendental numbers cannot. His diagonal argument, first presented in 1891, implies an unending hierarchy of infinities. Cantor, a devout Lutheran, believed God had communicated the theory to him, though some neo-Scholastic theologians saw it as a threat to God's unique absolute infinity.
Resistance was intense. Kronecker's opposition blocked Cantor's hopes of a Berlin chair, Henri Poincaré objected, and later Wittgenstein dismissed set theory as utter nonsense. Every one of the fifty-two letters Cantor sent Gösta Mittag-Leffler in 1884 mentioned Kronecker, and that May he suffered his first known bout of depression, which recurred for the rest of his life. For a time he tried lecturing on philosophy instead and wrote two pamphlets arguing Francis Bacon wrote Shakespeare's plays. The sudden death of his youngest son, Rudolph, after an 1899 hospitalisation drained much of his passion for mathematics.
Recognition came anyway. Cantor helped found the German Mathematical Society in 1889 and became its first president, even inviting Kronecker to speak, and the pair later reconciled. In 1904 the Royal Society gave him its Sylvester Medal, its top award for mathematics. He died on 6 January 1918.
Source: Georg Cantor
