, if not particularly helpful, way of envisaging infinity. “I generally think of it as: (a) very big, but (b) bigger than that,” says the mathematician from the University of Warwick in the UK. “When something is infinite, there is always some spare room around to put things in.”
Infinity is one of those things with a preprogrammed boggle factor. Mathematically, it started off as a way of expressing the fact that some things, like counting, have no obvious end. Count to 146 and there’s 147; count to a trillion and say hello to a trillion and one. There are two ways of dealing with this, says Stewart. “You can sum it up boldly as ‘there are infinitely many numbers’. But if you want to be more cautious, you just say ‘there is no largest number’.”
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Only in the late 19th century did mathematicians plump for the first option, and begin to handle infinity as an object with properties all its own. The key was set theory, a new way of thinking of numbers as bundles of things. The set of all whole numbers, for example, is a well-defined and unique object, and it has a size: infinity.
The sting in the tail, as the German mathematician Georg Cantor showed, is that by this definition there is more than one infinity. The set of the whole numbers defines one low-lying sort, known as countable infinity. But add in all the numbers in between, with as many decimal places as you please, and you get a smoother, more continuous infinity – one defined by a set that is infinitely bigger.
That is just the beginning. is a set theorist at Harvard University who has a whole level of infinity named after him, a particularly vertiginous level populated with numbers known as Woodin cardinals. “They are so large you can’t deduce their existence,” he says.
Such infinities help solve otherwise unsolvable problems in less rarefied mathematical landscapes below. They are the ultimate abstraction: although you can manipulate them logically, you can’t write formulae incorporating them or devise computer programs to test predictions about them. Woodin’s notepads consist mainly of cryptic marks he uses to focus his attention, to the occasional consternation of fellow plane passengers. “If they don’t try to change seats they ask me if I’m an artist,” he says.
How closely our common-sense conception of endlessness matches the mathematical infinities isn’t clear. But if we can’t quite grasp boundarylessness, it probably doesn’t matter, says Woodin – however you slice it, infinity seems far removed from anything we see in the real world. Perhaps those enigmatic markings aren’t so different from those of his fellow passengers after all. “It might be we’re just playing a game,” says Woodin. “Perhaps we are just doing some glorified sudoku puzzle.”
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This article appeared in print under the headline “How to think about infinity”
