infinity
/ in-FIN-uh-tee /
Infinity is endlessness — the idea of something that never stops, never runs out, has no last step. The crucial thing to get straight is that it is not a number. You can't reach it by counting, and you can't do ordinary arithmetic with it: there is no biggest number that counting finally arrives at, because whatever number you name, you can always add one more. Think of walking down a road that has no end — not a very long road, but one with genuinely no final mile.
Why it matters: Georg Cantor discovered something astonishing — there isn't just one infinity. Some infinities are strictly bigger than others. In 1874 he proved that the counting numbers 1, 2, 3, ... and all the real numbers between 0 and 1 are infinities of different sizes: the counting numbers go on forever, yet the points on that little stretch of line — which include the never-ending, never-repeating decimals, not just the tidy ones — form a crowd so much larger that they can't be paired off against the counting numbers at all. In 1891 he gave the elegant proof now usually told, the diagonal argument: line those numbers up in any list you like, then build a new one whose first digit differs from the first number's first digit, whose second digit differs from the second number's second digit, and so on (steering clear of 0s and 9s, so the same value can't sneak back in under a different decimal spelling). The number you get differs from every number on the list — so no list could ever hold them all.
A common confusion to clear up: there's the potential infinity of a process that can always go one more step (the ∞ symbol you meet in calculus, meaning "grow without bound"), and the actual infinity of a completed endless collection, like the whole set of counting numbers taken at once. Mathematicians once thought only the first kind was respectable; Cantor's great move was to treat the second kind as a real, working object — and to show it comes in different sizes.
The counting numbers and the even numbers are the SAME size — pair each n with 2n and none is left over, even though the evens are "half" of them.
With infinite sets, a part can be just as big as the whole — the test for "same size" is whether they can be paired off one-to-one.
The lemniscate symbol ∞ was introduced by the English mathematician John Wallis in 1655. Cantor later named the size of an endless set its "cardinality": the counting numbers have the smallest infinite cardinality, written ℵ₀ (aleph-null), while the real numbers have a strictly larger one.