Zero - Charles Seife [57]
Yielding a new number, .27800…, that
is different from the first number (their first digits don’t match),
is different from the second number (their second digits don’t match),
is different from the third number (their third digits don’t match),
is different from the fourth number (their fourth digits don’t match), and so forth.
Going down the diagonal in this way, we create a new number. This process ensures that it’s different from all the other numbers on the list. If it is different from all the numbers on the list, it can’t be on the list—but we already assumed our list contains all real numbers; after all, it was a perfect seating list. This is a contradiction. The perfect seating list cannot exist.
The real numbers are a bigger infinity than the rational numbers. The term for this type of infinity was , the first uncountable infinity. (Technically, the term for the infinity of the real line was C, or the continuum infinity. For years mathematicians struggled to determine whether C was indeed . In 1963 a mathematician, Paul Cohen, proved that this puzzle, the so-called continuum hypothesis, was neither provable nor disprovable, thanks to Gödel’s incompleteness theorem. Today most mathematicians accept the continuum hypothesis as true, though some study non-Cantorian transfinite numbers where the continuum hypothesis is taken to be false.) In Cantor’s mind there were an infinite number of infinities—the transfinite numbers—each nested in the other. is smaller than , which is smaller than , which is smaller than , and so forth. At the top of the chain sits the ultimate infinity that engulfs all other infinities: God, the infinity that defies all comprehension.
Unfortunately for Cantor, not everyone had the same vision of God. Leopold Kronecker was an eminent professor at the University of Berlin, and one of Cantor’s teachers. Kronecker believed that God would never allow such ugliness as the irrationals, much less an ever-increasing set of Russian-doll infinities. The integers represented the purity of God, while the irrationals and other bizarre sets of numbers were abominations—figments of the imperfect human mind. Cantor’s transfinite numbers were the worst of the lot.
Disgusted with Cantor, Kronecker launched vitriolic attacks against Cantor’s work and made it extremely difficult for him to publish papers. When Cantor applied for a position at the University of Berlin in 1883, he was rejected; he had to settle for a professorship at the much less prestigious University of Halle instead. Kronecker, who was influential at Berlin, was likely to blame. The same year, he wrote a defense against Kronecker’s attacks. Then, in 1884, the depressed Cantor had his first mental breakdown.
It would be little comfort to Cantor that his work was the foundation of a whole new branch of mathematics: set theory. Using set theory, mathematicians would not only create the numbers we know out of nothing at all, they would create numbers that were previously unheard of—infinite infinities that can be added to, multiplied with, subtracted from, and divided by other infinities, just like ordinary numbers. Cantor opened up a whole new universe of numbers. The German mathematician David Hilbert would say, “No one shall expel us from the paradise which Cantor has created for us.” But it was too late for Cantor. Cantor was in and out of mental institutions for the remainder of his life, and he died in the mental hospital at Halle in 1918.
In the battle between Kronecker and Cantor, Cantor would ultimately prevail. Cantor’s theory would show that Kronecker’s precious integers—and even the rational numbers—were nothing at all. They were an infinite zero.
There are an infinite number of rationals, and between any two numbers you choose, no matter how close together, there are still an infinite number of rationals. They are everywhere. But Cantor’s hierarchy of infinities would tell a different tale: it would show just how little space the rational numbers take up