To Infinity and Beyond
Illustration by Dana Collins 
TO MOST OF US, INFINITY LOOMS AS THE END OF the mathematical affair, the final and ineffable conclusion of a process that begins with the articulation of one, two, three. Infinity is the number that cannot be enumerated, the enigma lurking in the ellipses . . . It comes as a shock to learn, then, that for mathematicians infinity has become a trifling figure, not the last, but merely the first of a whole new class of numbers each more infinite than the one before. Of all the artifacts of the mathematical imagination, none is more fantastical than the transfinite numbers, the swollen and monstrous fruits of the counting spirit carried to its pathological extreme.
It has not always been thus. For most of history, philosophers and mathematicians alike recoiled against the infinite. The apeiron, Aristotle called it, declaring infinities of any variety an abomination. Christians, for the most part, agreed, though it is God who ultimately opened the door to infinity. At the tail end of the Middle Ages, the great early champion of mathematical science, Cardinal Nicholas of Cusa, first envisioned the possibility of infinite space. Though Cusa was careful to stress that only God Himself could be truly infinite, as God's creation he believed the universe should approximate the maximal spirit of its creator. Not an actual infinity, then, but at least a potential one.
Few mathematicians were inclined to follow. Not until the late 19th century would anyone wholly embrace the infinite, and even then the man who dared Georg Cantor faced almost universal hostility from his mathematical peers. Cantor's encounter with infinity ultimately drove him mad; like a 19thcentury counterpart to A Beautiful Mind's John Nash, he paid the price of his sanity for his unorthodox brand of genius.
It takes a special kind of courage to approach the Himalayan Alps of infinity even a cursory glance at the terrain is terrifying. As early as the ninth century, the Islamic mathematician Thabit ibn Qurra pointed out the paradoxes lurking within infinity, which unlike any other number can be split into multiple parts, each in itself as large as the whole. Consider the simplest kinds of numbers, one, two, three, four. Mathematicians call these the natural numbers, and as far as we can tell they go on forever. Yet the naturals can be split into two distinct groups, the odds and evens, both of which are also endless. Indeed, Cantor proved that the number of evens, or odds, is exactly the same as the total number of the naturals. Half of infinity still equals infinity! No other number behaves this way think of six, for example: Split it in half and you get three. Or eight: Halve that and you get four. But infinity operates by its own unique rules.
And it doesn't get easier. By the same method, Cantor showed that you can split the naturals into any number of subdivisions two, three, four, or 4 million and every one of these subgroups will have the same infinite number of members. If you are feeling somewhat uneasy at this point, it may help to know that many mathematicians feel the same way. All these infinities are equal, but Cantor wondered whether there could be a truly larger infinity? One obvious candidate would seem to be the fractions. Common sense surely dictates that there must be more fractions than natural numbers. Merely between the first two naturals, 0 and 1, there are already an infinite number of fractions 1/2, 1/3, 1/4, 1/5, 1/6. And so on. To most mathematicians the answer was selfevident and in no need of investigation. But to their horror, Cantor produced an ingenious proof which demonstrated that the number of fractions is precisely equal to the number of the naturals! Though the logic is irrefutable, and remarkably simple at that, it is hard to believe this result can really be true, so counter to common sense does it run.
Having scaled the peak of the fractions only to find himself back on the same ground from which he started, Cantor pressed on. The next obvious case was that of the socalled real numbers. This is the set of all the naturals and the fractions, with the addition of the irrational numbers, those that cannot be expressed as any kind of fraction. The irrationals include famous members like pi and phi (the golden mean) and have in common the fact that their decimal expansion goes on forever. Surely this class must be more infinite? Here Cantor was not disappointed; at last he had found a genuinely greater multitude.
Cantor sensed that ahead lay a vast terrain of ever more infinite infinities. "Mathematics is freedom!" he declared. In a moment of revelation came the thought that the infinity of the naturals what he called omega, the final letter of the Greek alphabet could itself be taken as a starting point. With the doors of perception thus unleashed, Cantor posited the existence of a whole new set of numbers beyond infinity, omega + 1, omega + 2, omega + 3 and so forth. Once begun, such a process is unstoppable. Into the distance stretches an endless sequence of omegas piled upon omegas, each begetting, like a biblical patriarch, new families of the infinite. These are the transfinites, and Cantor devoted the rest of his life to cataloging their properties. Just as the infinity of the real numbers is greater than the infinity of the naturals, so each new order of omegas gives rise to ever greater rankings of infinity, what mathematicians refer to as a greater kind of cardinal number. Cantor in his fervor named them simply the alephs, this time referencing the first letter of the Hebrew alphabet.
BUT IF THE AIR IS PURER AT HIGHER ALTITUDES, it is also thinner. As Cantor ascended into this rarefied realm, he began to suffer a series of mental breakdowns. In the touching biography, The Mystery of the Aleph, Amir Aczel recounts how Cantor was in and out of sanitariums, landing with increasing frequency in his university's nervenklinik. It is still a matter of debate whether it was the mathematics that drove him mad or the madness that drove him to pursue the math. Certainly he had plenty of colleagues who thought the very concept of transfinite numbers appalling. Leopold Kronecker, the most powerful mathematician in Germany at the time (and Germany was then the leading mathematical nation), repeatedly tried to prevent the publication of Cantor's papers, condemning him to a life of provincial isolation.
Cantor took this rejection as a sign of conspiracy. As with John Nash, the great brain began to wander. He started to concern himself with Theosophy, Freemasonry and the Rosicrucians; he attempted to prove that Shakespeare's plays had been written by Francis Bacon. The nervenklinik became a second home. "When you look long into an abyss," Nietzsche tells us, "the abyss also looks into you." Few men of any stripe have stared as long as Cantor.
During his life, Cantor remained a marginal force, but since his death his stature as a mathematician has, like his beloved transfinites, only continued to swell. Today he is enshrined in the pantheon of mathematical giants. What is more, the study of transfinite numbers has itself exceeded all limits. Harvard mathematician Robert Kaplan is coauthor of a forthcoming book, The Art of the Infinite, with his linguist wife, Ellen. Kaplan notes that today the transfinites are the center of an active research program which has led "to the discovery of numbers that dwarf even Cantor's remotest dreams." These peculiar "numbers," it must be said, bear little relationship to the common conception of that term; we are here in a realm of such arcane abstraction that some mathematicians themselves continue to question the validity of these entities.
The classes of infinities now under study sound indeed like refugees from Lewis Carroll's Wonderland remade by the Monty Python team: There is the First Inaccessible Cardinal and, after it, the Hyperinaccessibles. Then comes the first Mahlo Cardinal, which, as the Kaplans wryly remark, "sounds more like a medieval grandee than any kind of number." There are Indescribable Cardinals as well as Huge, Supercompact and Rowbottom Cardinals, to say nothing of the Extendible and Ineffable Cardinals. Last but not least are the Inexpressible Cardinals. "Devising them isn't only a game of oneupmanship on a gigantic scale," the Kaplans write, "but a serious attempt to prove important theories which are unprovable."
From Cantor's work inevitably followed the revolutionary theories of Kurt Goedel, which have changed forever our understanding of the foundation of rational thought. As Goedel demonstrated, every logical system inevitably contains paradoxes, propositions that can be proved both true and false. At the start of the 19th century, John Playfair noted the ineluctably creative dimension of modern science, an activity whose primary virtue is not necessarily empirical. By following the lead of rational conjecture, Playfair mused, we become aware "how much further reason may sometimes go than the imagination may dare to follow."
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