Infinity

• 1. Introduction

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  Infinity is something boundless, limitless, or endless, written with a symbol shaped like a sideways figure eight. For more than two thousand years, that simple idea refused to sit still. The ancient Greeks argued about whether it could even exist. In the 17th century, mathematicians started using infinite series and quantities they called infinitely small, yet nobody could say whether infinity was a number or merely an endless process. A tortoise with a head start once seemed to prove that a fast runner could never catch it. A Jain text sorted the infinite into three distinct orders. One mathematician decided the largest infinity was God. How did a word meaning unbounded become a precise tool that mathematicians manipulate like any other object? And why do cosmologists still not know whether the universe itself is infinite?

• 2. The Greek Horror

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  Anaximander, a pre-Socratic philosopher who lived around 610 to 546 BC, left the earliest recorded Greek idea of infinity. He used the word apeiron, meaning unbounded or indefinite. Aristotle, writing around 350 BC, split the concept in two: potential infinity, which he allowed, and actual infinity, which he rejected as the source of too many paradoxes. Some scholars argue the Hellenistic Greeks carried a horror of the infinite. The evidence sits inside Euclid, who around 300 BC never wrote that there are infinitely many primes. Instead he wrote that prime numbers are more than any assigned multitude of prime numbers. Yet others say Euclid, by proving the primes endless, was the first to overcome that very horror. Even his parallel postulate is contested. One translation has two straight lines, produced to infinity, meeting on a side; another softens it to produced indefinitely, dodging the word entirely. A few scholars go further and claim reflection on infinity actually underlay all early Greek philosophy, making Aristotle's caution the odd exception rather than the rule.

• 3. Achilles And The Tortoise

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  Zeno of Elea, who lived around 495 to 430 BC, never argued anything about the infinite directly. His paradoxes did the arguing for him, and Bertrand Russell called them immeasurably subtle and profound. In the most famous, Achilles races a tortoise and grants it a head start. Achilles runs to the tortoise's starting point, but by then the tortoise has moved on. He runs to that new spot, and again the tortoise has crept ahead. Step after step, the gap shrinks but never closes, so Achilles seems never to overtake the animal. Zeno belonged to the Eleatic school, which held that motion itself was an illusion. To him the real error was assuming Achilles could run at all. Thinkers who found that answer unacceptable spent over two millennia hunting for other flaws. The hunt ended in 1821, when Augustin-Louis Cauchy supplied a satisfactory definition of a limit. Give Achilles ten meters per second, the tortoise a tenth of that, and a hundred-meter head start, and Cauchy's pattern shows he does catch up.

• 4. Counting The Uncountable

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  Georg Cantor, near the end of the 19th century, did something no Greek had dared: he treated infinite numbers as objects to be measured against one another. He showed infinite sets come in different sizes. The cardinality of a line, seen as the set of all its points, is larger than the number of integers. Cantor named two kinds of infinite number. Ordinal numbers describe well-ordered sets, counting carried past any finite stopping point. Cardinal numbers describe how many members a set holds. The smallest infinite cardinal is aleph-null, the size of the natural numbers, and any set matching that size is countably infinite. A set too large to pair off with the positive integers is called uncountable. Cantor's theorem proved there is no largest cardinal at all, since every infinite set has a larger one. He called the collection of all sets an unincreasable absolute infinity, which he identified with God. His ideas drew controversy at first, but they prevailed, and modern mathematics now accepts actual infinity as part of a coherent theory. Richard Dedekind sharpened the foundation by defining an infinite set as one the same size as a strictly smaller subset of itself.

• 5. The Continuum

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  One of Cantor's most important results was that the real numbers outnumber the natural numbers. There are simply more points on the continuum than there are counting numbers. The continuum hypothesis asks whether any cardinal sits between those two sizes, and it claims none does. That hypothesis can be neither proved nor disproved within Zermelo-Fraenkel set theory, even when the Axiom of Choice is granted. Cardinal arithmetic delivers stranger results still. The number of points on a whole real line equals the number on any segment of it, and equals the number of points on a plane, and in any finite-dimensional space. The tangent function makes the first claim visible by pairing an interval with the entire line. The second Cantor proved in 1878, though it only became intuitive in 1890, when Giuseppe Peano introduced space-filling curves. These curves twist enough to fill an entire square, cube, or hypercube, matching the points on one side of a square to every point inside it.

• 6. Infinity In Practice

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  In real analysis the infinity symbol marks an unbounded limit, and it is not a real number. It records a quantity increasing without bound, or a series whose partial sums climb forever. Add the symbol as an actual value and the real numbers gain new structure: a two-point compactification, an extended real number system, or a one-point compactification known as the real projective line. Complex analysis treats infinity differently, as an unsigned limit where a magnitude grows past any value. A single point at infinity turns the complex plane into the Riemann sphere, and there division by zero finally becomes legal for any nonzero number. Gottfried Leibniz, a co-inventor of infinitesimal calculus, saw both infinitesimals and infinite quantities as ideal entities governed by his Law of continuity. In the second half of the 20th century, his infinitesimals were put on rigorous footing through nonstandard analysis. There an infinite number H yields distinct cousins: H plus H equals 2H, while H plus 1 stands apart, none of them equivalent the way Cantor's transfinites are.

• 7. Symbols And Skeptics

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  John Wallis introduced the infinity symbol in 1655, the same year he first used a notation for an infinite number in his De sectionibus conicis. The symbol, sometimes called the lemniscate, is encoded in Unicode and written in LaTeX as backslash infty. Since Wallis it has wandered beyond mathematics into modern mysticism and literary symbology. In his Arithmetica infinitorum of 1656, Wallis signaled infinite series and products by jotting a few terms and tacking on the abbreviation et cetera. Isaac Newton wrote about equations with infinitely many terms in 1699. Not everyone welcomed the trend. Leopold Kronecker, in the 1870s and 1880s, distrusted how his fellow mathematicians used infinity, and his skepticism grew into finitism, an extreme stance within constructivism and intuitionism. The Jain text Surya Prajnapti, from around the 4th to 3rd century BCE, had already sorted all numbers into enumerable, innumerable, and infinite, each split further into three orders, ending with nearly infinite, truly infinite, and infinitely infinite.

• 8. Does The Universe End

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  Thomas Digges in 1576 published the first proposal that the universe is infinite. Eight years later, in 1584, Giordano Bruno argued for an unbounded cosmos in On the Infinite Universe and Worlds, writing that innumerable suns exist and innumerable earths revolve around them, inhabited by living beings. Cosmologists still ask whether there are infinitely many stars, whether the universe has infinite volume, and whether space goes on forever. Being infinite is logically separate from having boundaries. The surface of the Earth is finite yet has no edge, since a straight path along its curvature eventually returns to its start, and the universe might share that topology. Measurements of multipole moments in the cosmic background radiation, recorded by the WMAP spacecraft, hint at a flat topology consistent with an infinite universe. Yet a flat universe can still be finite, like a video game where an object leaving one edge of the screen reappears on the other, a shape that is toroidal and flat at once. The idea reaches further into the multiverse hypothesis, which astrophysicist Michio Kaku describes as an infinite number and variety of universes, and into cyclic models positing an endless succession of Big Bangs.