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 Cantor’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. In a famous lecture presented in 1900 at the International Congress of Mathematicians in Paris, Hilbert placed this assertion, called the continuum hypothesis, at the top of a list of the 23 most important mathematics problems of the new century.
Proving the truth or falsehood of Cantor’s continuum hypothesis boils down to answering this: Where does the set of real numbers sit in the hierarchy of infinite sets? Is it really the very first uncountable set? Cantor, for one, suspected the answer was no, but he couldn’t prove it.
Cantor’s hierarchy of infinities was such a revolutionary concept that many of his contemporaries rejected it out of hand. Their derision, coupled with Cantor’s inability to prove the continuum hypothesis, sent him into several nervous breakdowns. At times, he gave up mathematics temporarily, opting to pursue instead another passion: trying to prove that the English philosopher Francis Bacon was the real author of William Shakespeare’s plays. In 1917, Cantor died, unhappy and depressed, in a sanatorium.
Declaration of independence
While Cantor was struggling with the continuum hypothesis, other mathematicians were exploring the implications of Cantor’s dramatically broad vision of sets. British mathematician Bertrand Russell demonstrated that if mathematicians were careless about how they defined sets, baffling paradoxes would result.
FLICKERING FIGURES. To prove that real numbers are uncountable, Georg Cantor invented his diagonal argument. He started with a square array of numbers where each row is different and then changed each number that falls on the diagonal. Reading the diagonal as a new row gives a number that’s not in the original array, so if there had been infinite rows, there would still be an additional possible row. Artist and mathematician Helaman Ferguson has created a visualization of that argument in this stereoscopic pair of images. The two square arrays of pixels are identical except for the diagonals (running from top left to bottom right), where a black pixel in one is a white pixel in the other. When viewed stereoscopically, the images combine, and the diagonal stands out from the background array, appearing to flicker. |
The elements of a set can be numbers, mathematical functions, or even sets themselves. Russell observed that a set conceivably could even contain itself, like a picture that includes a picture of itself, which includes a picture of itself, and so on. Russell asked mathematicians to consider a set—call it S—that contains all sets that do not contain themselves. Then he asked, Does S contain itself? Chase the cycle of implications around, and you’ll find that an answer of either yes or no leads to a logical contradiction.
To deal with this problem, mathematicians in the first quarter of the 20th century developed basic principles, or axioms, spelling out how sets behave and ruling out paradoxical sets such as Russell’s set S. These axioms are statements so natural and intuitive that mathematicians are willing to accept them without proof. One such axiom, for instance, states that given two sets, their elements can be collected together to make a new set. Another states that infinite sets exist. Despite their simplicity, these axioms—called the standard axioms of set theory—have enabled mathematicians to set up a rigorous framework for proving results in all mathematical fields, from fractals to differential equations.
The continuum hypothesis, however, exposed a glaring incompleteness in these axioms. In 1938, logician Kurt Gödel proved that the continuum hypothesis is consistent with the standard axioms of set theory. Then in 1963, Paul Cohen, now at Stanford University, proved that the opposite of the continuum hypothesis—the assertion that there is actually an infinite set that is bigger than the set of counting numbers but smaller than the set of real numbers—is also consistent with the axioms.
Put together, those two results indicate that it’s impossible either to prove or to disprove the continuum hypothesis using the standard axioms. This made many mathematicians conclude that it might never be possible to develop a satisfying sense of whether the hypothesis is true or false.
Cohen’s demonstration that the continuum hypothesis could be neither proved nor disproved “caused a foundational crisis,” Woodin says. “Here, we had a question which should have an answer, but it had been proven that there were no means of answering it.” This left mathematicians with a fundamental question: Does it even make sense to say the continuum hypothesis is true or false?
When it comes to the philosophical issue of the nature of truth, most mathematicians fall into one of two camps called formalism and Platonism. Formalists take the position that mathematical statements don’t have an intrinsic truth or falsity—that the only thing that can ever be said about a statement is whether it can be proved in a given axiom system.
To formalists, it makes no sense to talk about whether the continuum hypothesis is true or false. They hold that, if the continuum hypothesis can’t be resolved within the standard framework of mathematics, then the hypothesis must be inherently vague. “Some people think it’s as intractable as asking how many angels can dance on the head of a pin, or what color is the number pi,” Woodin says.
To Platonists, mathematical objects such as sets exist in an ideal mathematical world, and axiomatic systems are merely useful tools for illuminating which statements about those objects are true in that world. To Platonists, the continuum hypothesis feels like a concrete statement that should be true or false. To them, if the standard axioms can’t settle the continuum hypothesis, it’s not that the hypothesis is a meaningless question, but rather that the axioms are insufficient.
From this point of view, Cohen’s result indicates that mathematicians need to add to their roster of axioms about infinite sets. There is a problem, however. An axiom should be so intuitively obvious that everyone agrees immediately that it’s true. Yet intuition quickly evaporates when confronted with questions about infinity.
Infinite elegance
In the decades that followed Cohen’s 1963 result, mathematicians trying to settle the continuum hypothesis ran into a roadblock: While some people proposed new axioms indicating the continuum hypothesis was true, others proposed what seemed like equally good axioms indicating the it false, Woodin says.
Woodin decided to try a different tack. Instead of looking for the missing axiom, he gathered circumstantial evidence about what the implications of that axiom would be. To do this without knowing what the axiom was, Woodin tried to figure out whether some axioms are somehow better than others. A good axiom, he felt, should help mathematicians settle not only the continuum hypothesis but also many other questions about Cantor’s hierarchy of infinite sets.
Mathematicians have long known that there is no all-powerful axiom that can answer every question about Cantor’s hierarchy. However, Woodin suspected a compromise is possible: There might be axioms that answer all questions up to the level of the hierarchy that the continuum hypothesis concerns—the realm of the smallest uncountably infinite sets. Woodin called such an axiom “elegant.”
In a book-length mathematical argument that has been percolating through the set theory community for the last few years, Woodin has proved—apart from one missing piece that must still be filled in—that elegant axioms do exist and, crucially, that every elegant axiom would make the continuum hypothesis false.
“If there’s a simple solution to the continuum hypothesis, it must be that it is false,” Woodin says. And if it is false, then there are indeed infinite sets bigger than the counting numbers and smaller than the real numbers.
Woodin’s novel approach of sidestepping the search for the right axiom doesn’t conform to the way mathematicians thought the continuum hypothesis would be settled, says Joel Hamkins of the City University of New York and Georgia State University in Atlanta.
Mathematicians haven’t yet absorbed the ramifications of Woodin’s work fully enough to decide whether it settles the matter of the continuum hypothesis, says Akihiro Kanamori of Boston University (Mass.). “[It’s] considered a very impressive achievement, but very few people understand the higher reaches [of Woodin’s framework],” he says.
Does Woodin himself believe that the continuum hypothesis is false? “If anyone should have an opinion on this, I should, but even I’m not sure,” he answers. “What I can say is that 10 years ago, I wouldn’t have believed there was a chance the continuum hypothesis was solvable. Now, I really think it has an answer.”
****************
If you have a comment on this article that you would like considered for publication in Science News, send it to editors@sciencenews.org. Please include your name and location.
