Infinite Wisdom: A New Approach to One of Mathematics' Most Notorious Problems

By Klarreich, Erica | Science News, August 30, 2003 | Go to article overview

Infinite Wisdom: A New Approach to One of Mathematics' Most Notorious Problems


Klarreich, Erica, Science News


How many numbers are there? For children, the answer might be a million--that is, until they discover a billion, or a trillion, or a googol. Then, maybe they notice that a googol plus one is also a number, and they realize that although the names for numbers run out, the numbers themselves never do. Yet to mathematicians, the idea that there are infinitely many numbers is just the beginning of an answer. Counterintuitive as it seems, there are many infinities--infinitely many, in fact. And some are bigger than others.

In the late 19th century, mathematicians showed that most familiar infinite collections of numbers are the same size. This group includes the counting numbers (1, 2, 3, ...), the even numbers, and the rational numbers (quotients of counting numbers, such as 3/4 and 101/763). However, in work that astonished the mathematicians of his day, the Russian-born Georg Cantor proved in 1873 that the real numbers (all the numbers that make up the number line) form a bigger infinity than the counting numbers do.

If that's the case, how much bigger is that infinity? This innocent-sounding question has stumped mathematicians from Cantor's time to the present. More than that, the question has exposed a gaping hole in the foundations of mathematics and has led mathematicians to reexamine the very nature of mathematical truth.

Now, Hugh Woodin, a mathematician at the University of California, Berkeley, may finally have found a way to resolve the issue, long considered one of the most fundamental in mathematics.

"It's a remarkable piece of mathematics," says Patrick Dehornoy of the University of Caen in France. He presented a lecture on Woodin's work last March at the Bourbaki seminar in Paris, one of the most famous and long-standing seminars in mathematics.

A PARADISE OF INFINITIES At first glance, it might seem obvious that the real numbers form a bigger infinity than the counting numbers do. After all, the real number line is an infinitely long, continuous expanse, while the counting numbers are just isolated milestones along this line.

However, little is obvious when it comes to infinite sets. For one thing, there's no way to simply count the elements of two infinite sets and determine which set has more. Instead, mathematicians say two infinite sets are the same size if there's a way to pair their elements, one to one, with no elements of either set left over.

Oddly enough, by this measure, the infinite set of counting numbers {1, 2, 3, ...} is the same size as the infinite set of even numbers {2, 4, 6, ...}, despite the fact that the even numbers make up precisely half of the counting numbers. The pairing procedure here works by hooking up 1 with 2, 2 with 4, 3 with 6, 4 with 8, and so on to make a perfect one-to-one correspondence between the two sets.

At first, mathematicians thought that all infinite sets could be paired with the counting numbers in this way. However, Cantor came up with an ingenious argument to show that there is no way to match the real numbers with the counting numbers without having real numbers left over. Because of this, mathematicians now refer to the infinite set of real numbers as uncountable.

Once Cantor had shown that the real numbers make up a bigger infinity than the counting numbers do, he saw no reason to stop there. He realized that there's an entire hierarchy of infinities--a "paradise of infinities," in the words of the great German mathematician David Hilbert, one of Cantot's contemporaries.

Cantor studied many infinite sets of numbers, but he never found one whose size fell between that of the counting numbers and the real numbers. So in 1877, he speculated that the real numbers, often called the continuum, form the smallest possible infinite set that is bigger than the counting numbers. In other words, there should be no set of numbers larger than the set of counting numbers but smaller than the set of real numbers. …

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Infinite Wisdom: A New Approach to One of Mathematics' Most Notorious Problems
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