Wall Street Journal

Definition of Infinity
Expands for Scientists
And Mathematicians
July 29, 2005; Page B1

At the Hotel Infinity, managers never have a problem with overbooking. If you arrive with a reservation and find that the hotel's infinite number of rooms (named 1, 2, 3 and so on, forever) are all occupied, the manager simply moves the guest in Room 1 to Room 2, the guest in Room 2 to Room 3, and on and on until every guest has a room and you get Room 1. In an "infinite set" such as the rooms at the Hotel, whatever you thought was the highest-numbered member of that set isn't.

The next time you're in town, you have an infinite number of friends in tow, and you try the Hotel Infinity again. The manager is happy to accommodate a party of infinity even though his infinite rooms are, again, full. Knowing that your friends have an odd aversion to even numbers, he moves the guest in Room 1 to Room 2, the guest in Room 2 to Room 4, the guest in Room 3 to Room 6, etc. You and your friends get the odd-numbered rooms, of which there are, conveniently, an infinite number.

If thinking of infinities makes your head spin, you're in good company. Georg Cantor, the early-20th-century mathematician who did more than anyone to explore infinities, suffered a nervous breakdown and repeated bouts of depression. In the 1930s, some fed-up mathematicians even argued that infinities should be banned from mathematics. Today, however, infinities aren't just a central part of mathematics. More surprising, says cosmologist John Barrow of the University of Cambridge, England, in his charming new tome, "The Infinite Book," scientists who study the real world are having to take infinities seriously, too.

Not long ago, if the solution to an equation included an infinity, alarms went off. In particle physics, for instance, "the appearance of an infinite answer was always taken as a warning that you had made a wrong turn," Prof. Barrow says. So physicists performed a sleight-of-hand, subtracting the infinite part of the answer and leaving the finite part. The finite part produced by this "renormalization" was always in "spectacularly good agreement with experiments," he says, but "there was always a deep uneasiness" over erasing infinities so blithely. Might physicists, blinded by their abhorrence of infinities, have been erasing a deep truth of nature?

Suspecting just that, some scientists now see infinities "as an essential part of the physical description of the universe," says Prof. Barrow. For instance, Einstein's equations say the universe began in, and will end with, an infinity of density and temperature, something long regarded as a sign that his theory breaks down at the beginning and end of time. But in a 2004 paper, Prof. Barrow calculated that Einstein's equations allow a point of infinite pressure to arise throughout the expanding universe at some time in the future.

In addition to coming around to the view that infinities might be real, rather than signs of a problem with Einstein's and other theories, some cosmologists suspect that infinities at the beginning and end of time "have quite different structures," Prof. Barrow writes. Just as at the hotel, not all infinities are equal. And that is making the weird math of different-size infinities suddenly relevant in the physical world, too.

To mathematicians, "equal" means you can match the elements in one set to the elements in another, one to one, with nothing left over. For instance, there is an infinite number of integers: 1, 2, 3, 4 . . . . There is also an infinite number of squares: 1, 4, 9, 16 . . . . You can match every integer with a square (1 with 1, 2 with 4, and so on), so the two sets are equal, as long as you never stop matching. But wait: Every square also belongs to the set of integers. That suggests that the set of integers is larger, since it contains all the squares and then some. Surely there are more integers than squares, right?

Actually, no. Before his breakdown, Cantor asserted that if the elements in one infinite set match up one to one with the counting numbers, then those infinities are of equal size. The infinity of squares and the infinity of integers (and the infinity of even numbers) are therefore equal, even though the infinity of integers is denser.

Decimals, however, are different, mathematicians say. There is an infinite number of them, too, but this infinity is larger than the infinity of integers or squares. Even in the tiny space between zero and 1, there's an infinite number of decimals with no certainty as to what comes next. What comes after .1, for example? Is it .11 or .2?

Just as mathematicians found a distinction among infinities, so scientists trying to fathom the physical world may need to distinguish among infinities.

In his study of infinities, Prof. Barrow noticed that a universe like ours that seems infinite in size, extending without bound, presents curious ethical dilemmas. An infinite universe must have infinite amounts of good and evil, he writes. Nothing we do, or fail to do, can change that, for adding a bit of good to an infinite amount of good still leaves infinite good, and subtracting a bit of evil from an infinite amount of evil still leaves infinite evil. "What is the status of good and evil," he wonders, "when all possible outcomes actually arise somewhere" ... or sometime? Small wonder infinity drove Cantor mad.