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INFINITE WISDOMA new approach to one of mathematics’

most notorious problemsBY ERICA KLARREICH

How many numbers are there? For children, theanswer might be a million—that is, until theydiscover a billion, or a trillion, or a googol.Then, maybe they notice that a googol plusone is also a number, and they realize that

although the names for numbers run out, the numbersthemselves never do. Yet to mathematicians, the ideathat there are infinitely many numbers is just the begin- ning of an answer. Counterintuitive as it seems, there are many infini-ties—infinitely many, in fact. And some are bigger than others.

In the late 19th century, mathematicians showed that most famil-iar infinite collections of numbers are the same size. This groupincludes the counting numbers (1, 2,3, . . .), the even numbers, and therational numbers (quotients of count-ing numbers, such as 3/4 and101/763). However, in work thatastonished the mathematicians of hisday, the Russian-born Georg Cantorproved in 1873 that the real numbers(all the numbers that make up thenumber line) form a bigger infinitythan the counting numbers do.

If that’s the case, how much biggeris that infinity? This innocent-sound-ing question has stumped mathe-maticians from Cantor’s time to thepresent. More than that, the questionhas exposed a gaping hole in the foun-dations of mathematics and has ledmathematicians to reexamine the verynature of mathematical truth.

Now, Hugh Woodin, a mathemati-cian at the University of California,Berkeley, may finally have found away to resolve the issue, long consid-ered one of the most fundamental inmathematics.

“It’s a remarkable piece of mathematics,” says Patrick Dehornoyof the University of Caen in France. He presented a lecture onWoodin’s work last March at the Bourbaki seminar in Paris, oneof the most famous and long-standing seminars in mathematics.

A PARADISE OF INFINITIES At first glance, it might seemobvious that the real numbers form a bigger infinity than the count-ing numbers do. After all, the real number line is an infinitely long,continuous expanse, while the counting numbers are just isolatedmilestones 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 infinitesets and determine which set has more. Instead, mathematicianssay two infinite sets are the same size if there’s a way to pair theirelements, one to one, with no elements of either set left over.

Oddly enough, by this measure, the infinite set of countingnumbers {1, 2, 3, . . . } is the same size as the infinite set of evennumbers {2, 4, 6, . . . }, despite the fact that the even numbersmake up precisely half of the counting numbers. The pairingprocedure here works by hooking up 1 with 2, 2 with 4, 3 with6, 4 with 8, and so on to make a perfect one-to-one correspon-dence between the two sets.

At first, mathematicians thought that all infinite sets could bepaired with the counting numbers in this way. However, Cantorcame up with an ingenious argument to show that there is no wayto match the real numbers with the counting numbers withouthaving real numbers left over. Because of this, mathematicians

now refer to the infinite set of realnumbers as uncountable.

Once Cantor had shown that thereal numbers make up a bigger infin-ity than the counting numbers do, hesaw no reason to stop there. He real-ized that there’s an entire hierarchy ofinfinities—a “paradise of infinities,” inthe words of the great German math-ematician David Hilbert, one of Can-tor’s contemporaries.

Cantor studied many infinite sets ofnumbers, but he never found onewhose size fell between that of thecounting numbers and the real num-bers. So in 1877, he speculated that thereal numbers, often called the contin-uum, form the smallest possible infi-nite set that is bigger than the count-ing numbers. In other words, thereshould be no set of numbers largerthan the set of counting numbers butsmaller than the set of real numbers.In a famous lecture presented in 1900at the International Congress of Math-ematicians in Paris, Hilbert placed this

assertion, called the continuum hypothesis, at the top of a list of the23 most important mathematics problems of the new century.

Proving the truth or falsehood of Cantor’s continuum hypothe-sis boils down to answering this: Where does the set of real num-bers sit in the hierarchy of infinite sets? Is it really the very firstuncountable set? Cantor, for one, suspected the answer was no, buthe couldn’t prove it.

Cantor’s hierarchy of infinities was such a revolutionary con-cept that many of his contemporaries rejected it out of hand. Theirderision, coupled with Cantor’s inability to prove the continuumhypothesis, sent him into several nervous breakdowns. At times,

INFINITE REGRESS — Architectural features, such asa ceiling that consists of stacked polygons of diminishingsize, can create an illusion of infinite extent.

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he gave up mathematics temporarily, opting to pursue insteadanother passion: trying to prove that the English philosopherFrancis 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 wasstruggling with the continuum hypothesis, other mathematicianswere exploring the implications of Cantor’s dramatically broadvision of sets. British mathematician Bertrand Russell demon-strated that if mathematicians were careless about how they definedsets, baffling paradoxes would result.

The elements of a set can be numbers, mathematical functions,or even sets themselves. Russell observed that a set conceivablycould even contain itself, like a picture that includes a picture ofitself, which includes a picture of itself, and so on. Russell askedmathematicians to consider a set—call it S—that contains all setsthat do not contain themselves. Then he asked, Does S containitself? Chase the cycle of implications around, and you’ll find thatan answer of either yes or no leads to a logical contradiction.

To deal with this problem, mathematicians in the first quarterof the 20th century developed basic principles, or axioms, spellingout how sets behave and ruling out paradoxical sets such as Rus-sell’s set S. These axioms are statements so natural and intuitivethat mathematicians are willing to accept them without proof.One such axiom, for instance, states that given two sets, their ele-ments can be collected together to make a new set. Another statesthat infinite sets exist. Despite their simplicity, these axioms—called the standard axioms of set theory—have enabled mathe-maticians to set up a rigorous framework for proving results in allmathematical fields, from fractals to differential equations.

The continuum hypothesis, however, exposed a glaring incom-pleteness in these axioms. In 1938, logician Kurt Gödel provedthat the continuum hypothesis is consistent with the standardaxioms of set theory. Then in 1963, Paul Cohen, now at StanfordUniversity, 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 num-bers—is also consistent with the axioms.

Put together, those two results indicate that it’s impossible eitherto prove or to disprove the continuum hypothesis using the stan-dard axioms. This made many mathematicians conclude that itmight 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 nei-ther proved nor disproved “causeda foundational crisis,” Woodin says.“Here, we had a question whichshould have an answer, but it hadbeen proven that there were nomeans of answering it.” This leftmathematicians with a fundamen-tal question: Does it even makesense to say the continuum hypoth-esis is true or false?

When it comes to the philosoph-ical issue of the nature of truth, mostmathematicians fall into one of two

camps called formalism and Platonism. Formalists take the posi-tion that mathematical statements don’t have an intrinsic truth orfalsity—that the only thing that can ever be said about a statementis whether it can be proved in a given axiom system.

To formalists, it makes no sense to talk about whether the con-tinuum hypothesis is true or false. They hold that, if the continuumhypothesis can’t be resolved within the standard framework ofmathematics, then the hypothesis must be inherently vague. “Somepeople think it’s as intractable as asking how many angels can danceon the head of a pin, or what color is the number pi,” Woodin says.

To Platonists, mathematical objects such as sets exist in an idealmathematical world, and axiomatic systems are merely usefultools for illuminating which statements about those objects aretrue in that world. To Platonists, the continuum hypothesis feels

“Here, we had aquestion whichshould have ananswer, but ithad been proventhat there wereno means ofanswering it.”—HUGH WOODIN

FLICKERING FIGURES — To prove that real numbers are uncountable, Georg Cantor invented his diagonal argument. He started with asquare array of numbers where each row is different and then changed each number that falls on the diagonal. Reading the diagonal as anew 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 squarearrays 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 inthe other. When viewed stereoscopically, the images combine, and the diagonal stands out from the background array, appearing to flicker.

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like a concrete statement that should be true or false. To them, ifthe standard axioms can’t settle the continuum hypothesis, it’snot that the hypothesis is a meaningless question, but rather thatthe axioms are insufficient.

From this point of view, Cohen’s result indi-cates that mathematicians need to add to theirroster of axioms about infinite sets. There is aproblem, however. An axiom should be so intu-itively obvious that everyone agrees immediatelythat it’s true. Yet intuition quickly evaporates whenconfronted with questions about infinity.

INFINITE ELEGANCE In the decades that followedCohen’s 1963 result, mathematicians trying to settle thecontinuum hypothesis ran into a roadblock: While somepeople proposed new axioms indicating the continuum hypoth-esis was true, others proposed what seemed like equally goodaxioms indicating the itfalse, Woodin says.

Woodin decided to trya different tack. Insteadof looking for the miss-ing axiom, he gatheredcircumstantial evidenceabout what the implica-tions of that axiom wouldbe. To do this withoutknowing what the axiomwas, Woodin tried to fig-ure out whether some axioms are somehow better than others. Agood axiom, he felt, should help mathematicians settle not only thecontinuum hypothesis but also many other questions about Can-tor’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 mightbe axioms that answer all questions up to the level of the hierarchythat the continuum hypothesis concerns—the realm of the smallestuncountably infinite sets. Woodin called such an axiom “elegant.”

In a book-length mathematical argument that has been percolatingthrough the set theory community for the last few years, Woodin hasproved—apart from one missing piece that must still be filled in—

that elegant axioms do exist and, crucially, that every elegantaxiom would make the continuum hypothesis false.

“If there’s a simple solution to the continuumhypothesis, it must be that it is false,” Woodin

says. And if it is false, then there areindeed infinite sets bigger than the

counting numbers and smallerthan the real numbers. Woodin’s novel approach of side-

stepping the search for the right axiomdoesn’t conform to the way mathematicians

thought the continuum hypothesis would be set-tled, says Joel Hamkins of the City University ofNew York and Georgia State University in Atlanta.

Mathematicians haven’t yet absorbed theramifications of Woodin’s work fully enough to

decide whether it settles the matter of the contin-uum hypothesis, says Akihiro Kanamori of Boston

University (Mass.). “[It’s] considered a very impressive achieve-ment, but very few people understand the higher reaches [ofWoodin’s framework],” he says.

Does Woodin himself believe that the continuum hypothesisis false? “If anyone should have an opinion on this, I should, buteven 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 hypoth-esis was solvable. Now, I really think it has an answer.” �H

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BRANCHING TO INFINITY —Remove the middle third of a linesegment, remove the middle third ofthe two remaining pieces, and thencontinue this process ad infinitum.The collection of points that remains,called a Cantor set, is an example of aninfinite set. Here, a branched visualizationof the process as a binary tree appears inthe work of art “Infinite Cactus.”

PALEOBIOLOGY

Oh, what a stickyweb they wove

A look inside a piece of 130-million-year-old amber found in Lebanon has revealeda gossamer treasure: a filament of spidersilk laced with sticky droplets that look justlike those from modern spiders.

The 4-millimeter-long strand of viscidsilk—the glue-covered type that some web-spinning spiders use to capture prey—ismore than 90 million years older than anyknown sample of spider silk.

Despite its age, the strand has hallmarksof modern spider silk, says SamuelZschokke, a biologist at the University ofBasel in Switzerland. For example, most of

the filament’s glue droplets range from 7 to29 micrometers in diameter and arearranged in an alternating sequence of smalland large. Zschokke describes the delicatefossil in the Aug. 7 Nature.

Both modern orb-weaver spiders andcomb-footed spiders spin this type of silk.If the fossil filament came from an ances-tor of one of those varieties, it was proba-bly a comb-footed spider, says Zschokke.Today, those arachnids are the only onesthat spin webs bearing viscid silk near treetrunks, where seeping resin would be likelyto trap a stray strand of silk. —S.P.

BEHAVIOR

Meditation changes

People who meditate say that the practicecalms them and improves their perform-ance on everyday tasks. There may be foun-dations of these benefits in the brain andimmune system, a new study finds.

Psychologist Richard J. Davidson of theUniversity of Wisconsin–Madison and his

colleagues studied 41 employees of abiotechnology company, 25 of whom com-pleted an 8-week meditation program.The scientists measured brain wave activ-ity in all participants before, immediatelyfollowing, and 4 months after the medi-tation program. Volunteers also receivedan influenza vaccination at the end of theprogram and gave blood samples 1 monthand 2 months later, enabling the research-ers to assess the volunteers’ immuneresponses to the vaccine.

Only the meditators exhibited increasesin brain wave activity across the front ofthe left hemisphere, Davidson’s groupreports in the July/August PsychosomaticMedicine. Earlier studies had suggestedthat this neural response accompaniesboth reductions in negative emotions andsurges in positive emotions. The employ-ees who took the course reported subse-quent drops in negative feelings but nochange in pleasant feelings.

Meditators displayed more-vigorousantibody responses to the vaccine than theirnonmeditating peers did. —B.B.

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