mathematicians measure infinities and find they’re equal · mathematicians subsequently used...

Quanta Magazine

https://www.quantamagazine.org/mathematicians-measure-infinities-find-theyre-equal-20170912/ September 12, 2017

Mathematicians Measure Infinities and FindThey’re EqualTwo mathematicians have proved that two different infinities are equal in size, settling a long-standing question. Their proof rests on a surprising link between the sizes of infinities and thecomplexity of mathematical theories.

By Kevin Hartnett

In a breakthrough that disproves decades of conventional wisdom, two mathematicians have shownthat two different variants of infinity are actually the same size. The advance touches on one of themost famous and intractable problems in mathematics: whether there exist infinities between theinfinite size of the natural numbers and the larger infinite size of the real numbers.

The problem was first identified over a century ago. At the time, mathematicians knew that “the realnumbers are bigger than the natural numbers, but not how much bigger. Is it the next biggest size,or is there a size in between?” said Maryanthe Malliaris of the University of Chicago, co-author ofthe new work along with Saharon Shelah of the Hebrew University of Jerusalem and RutgersUniversity.

In their new work, Malliaris and Shelah resolve a related 70-year-old question about whether oneinfinity (call it p) is smaller than another infinity (call it t). They proved the two are in fact equal,much to the surprise of mathematicians.

“It was certainly my opinion, and the general opinion, that p should be less than t,” Shelah said.

Malliaris and Shelah published their proof last year in the Journal of the American MathematicalSociety and were honored this past July with one of the top prizes in the field of set theory. But theirwork has ramifications far beyond the specific question of how those two infinities are related. Itopens an unexpected link between the sizes of infinite sets and a parallel effort to map thecomplexity of mathematical theories.

Many InfinitiesThe notion of infinity is mind-bending. But the idea that there can be different sizes of infinity?That’s perhaps the most counterintuitive mathematical discovery ever made. It emerges, however,from a matching game even kids could understand.

Suppose you have two groups of objects, or two “sets,” as mathematicians would call them: a set ofcars and a set of drivers. If there is exactly one driver for each car, with no empty cars and nodrivers left behind, then you know that the number of cars equals the number of drivers (even if you

http://www.ams.org/journals/jams/2016-29-01/S0894-0347-2015-00830-X/home.html

https://ests.wordpress.com/2017/07/05/third-hausdorff-medal-2017/

Quanta Magazine


don’t know what that number is).

In the late 19th century, the German mathematician Georg Cantor captured the spirit of thismatching strategy in the formal language of mathematics. He proved that two sets have the samesize, or “cardinality,” when they can be put into one-to-one correspondence with each other — whenthere is exactly one driver for every car. Perhaps more surprisingly, he showed that this approachworks for infinitely large sets as well.

Consider the natural numbers: 1, 2, 3 and so on. The set of natural numbers is infinite. But whatabout the set of just the even numbers, or just the prime numbers? Each of these sets would at firstseem to be a smaller subset of the natural numbers. And indeed, over any finite stretch of thenumber line, there are about half as many even numbers as natural numbers, and still fewer primes.

Yet infinite sets behave differently. Cantor showed that there’s a one-to-one correspondencebetween the elements of each of these infinite sets.

1 2 3 4 5 … (natural numbers)2 4 6 8 10 … (evens)2 3 5 7 11 … (primes)

Because of this, Cantor concluded that all three sets are the same size. Mathematicians call sets ofthis size “countable,” because you can assign one counting number to each element in each set.

After he established that the sizes of infinite sets can be compared by putting them into one-to-onecorrespondence with each other, Cantor made an even bigger leap: He proved that some infinite setsare even larger than the set of natural numbers.

Quanta Magazine


Quanta Magazine


Lucy Reading-Ikkanda/Quanta Magazine

Consider the real numbers, which are all the points on the number line. The real numbers aresometimes referred to as the “continuum,” reflecting their continuous nature: There’s no spacebetween one real number and the next. Cantor was able to show that the real numbers can’t be putinto a one-to-one correspondence with the natural numbers: Even after you create an infinite listpairing natural numbers with real numbers, it’s always possible to come up with another realnumber that’s not on your list. Because of this, he concluded that the set of real numbers is largerthan the set of natural numbers. Thus, a second kind of infinity was born: the uncountably infinite.

What Cantor couldn’t figure out was whether there exists an intermediate size of infinity —something between the size of the countable natural numbers and the uncountable real numbers. Heguessed not, a conjecture now known as the continuum hypothesis.

In 1900, the German mathematician David Hilbert made a list of 23 of the most important problemsin mathematics. He put the continuum hypothesis at the top. “It seemed like an obviously urgentquestion to answer,” Malliaris said.

In the century since, the question has proved itself to be almost uniquely resistant tomathematicians’ best efforts. Do in-between infinities exist? We may never know.

Forced OutThroughout the first half of the 20th century, mathematicians tried to resolve the continuumhypothesis by studying various infinite sets that appeared in many areas of mathematics. They hopedthat by comparing these infinities, they might start to understand the possibly non-empty spacebetween the size of the natural numbers and the size of the real numbers.

Many of the comparisons proved to be hard to draw. In the 1960s, the mathematician Paul Cohenexplained why. Cohen developed a method called “forcing” that demonstrated that the continuumhypothesis is independent of the axioms of mathematics — that is, it couldn’t be proved within theframework of set theory. (Cohen’s work complemented work by Kurt Gödel in 1940 that showed thatthe continuum hypothesis couldn’t be disproved within the usual axioms of mathematics.)

Cohen’s work won him the Fields Medal (one of math’s highest honors) in 1966. Mathematicianssubsequently used forcing to resolve many of the comparisons between infinities that had beenposed over the previous half-century, showing that these too could not be answered within theframework of set theory. (Specifically, Zermelo-Fraenkel set theory plus the axiom of choice.)

Some problems remained, though, including a question from the 1940s about whether p is equal tot. Both p and t are orders of infinity that quantify the minimum size of collections of subsets of thenatural numbers in precise (and seemingly unique) ways.

The details of the two sizes don’t much matter. What’s more important is that mathematiciansquickly figured out two things about the sizes of p and t. First, both sets are larger than the naturalnumbers. Second, p is always less than or equal to t. Therefore, if p is less than t, then p would bean intermediate infinity — something between the size of the natural numbers and the size of thereal numbers. The continuum hypothesis would be false.

Mathematicians tended to assume that the relationship between p and t couldn’t be proved within

Quanta Magazine


the framework of set theory, but they couldn’t establish the independence of the problem either. Therelationship between p and t remained in this undetermined state for decades. When Malliaris andShelah found a way to solve it, it was only because they were looking for something else.

An Order of ComplexityAround the same time that Paul Cohen was forcing the continuum hypothesis beyond the reach ofmathematics, a very different line of work was getting under way in the field of model theory.

Quanta Magazine


Courtesy of H. Jerome Keisler

H. Jerome Keisler invented “Keisler’s order.”

Quanta Magazine


For a model theorist, a “theory” is the set of axioms, or rules, that define an area of mathematics.You can think of model theory as a way to classify mathematical theories — an exploration of thesource code of mathematics. “I think the reason people are interested in classifying theories is theywant to understand what is really causing certain things to happen in very different areas ofmathematics,” said H. Jerome Keisler, emeritus professor of mathematics at the University ofWisconsin, Madison.

In 1967, Keisler introduced what’s now called Keisler’s order, which seeks to classify mathematicaltheories on the basis of their complexity. He proposed a technique for measuring complexity andmanaged to prove that mathematical theories can be sorted into at least two classes: those that areminimally complex and those that are maximally complex. “It was a small starting point, but myfeeling at that point was there would be infinitely many classes,” Keisler said.

It isn’t always obvious what it means for a theory to be complex. Much work in the field is motivatedin part by a desire to understand that question. Keisler describes complexity as the range of thingsthat can happen in a theory — and theories where more things can happen are more complex thantheories where fewer things can happen.

A little more than a decade after Keisler introduced his order, Shelah published an influential book,which included an important chapter showing that there are naturally occurring jumps in complexity— dividing lines that distinguish more complex theories from less complex ones. After that, littleprogress was made on Keisler’s order for 30 years.

Quanta Magazine


Yael Shelah

Saharon Shelah is a co-author of the new proof.

Quanta Magazine


Then, in her 2009 doctoral thesis and other early papers, Malliaris reopened the work on Keisler’sorder and provided new evidence for its power as a classification program. In 2011, she and Shelahstarted working together to better understand the structure of the order. One of their goals was toidentify more of the properties that make a theory maximally complex according to Keisler’scriterion.

Malliaris and Shelah eyed two properties in particular. They already knew that the first one causesmaximal complexity. They wanted to know whether the second one did as well. As their workprogressed, they realized that this question was parallel to the question of whether p and t areequal. In 2016, Malliaris and Shelah published a 60-page paper that solved both problems: Theyproved that the two properties are equally complex (they both cause maximal complexity), and theyproved that p equals t.

“Somehow everything lined up,” Malliaris said. “It’s a constellation of things that got solved.”

This past July, Malliaris and Shelah were awarded the Hausdorff medal, one of the top prizes in settheory. The honor reflects the surprising, and surprisingly powerful, nature of their proof. Mostmathematicians had expected that p was less than t, and that a proof of that inequality would beimpossible within the framework of set theory. Malliaris and Shelah proved that the two infinitiesare equal. Their work also revealed that the relationship between p and t has much more depth to itthan mathematicians had realized.

“I think people thought that if by chance the two cardinals were provably equal, the proof wouldmaybe be surprising, but it would be some short, clever argument that doesn’t involve building anyreal machinery,” said Justin Moore, a mathematician at Cornell University who has published a briefoverview of Malliaris and Shelah’s proof.

Instead, Malliaris and Shelah proved that p and t are equal by cutting a path between model theoryand set theory that is already opening new frontiers of research in both fields. Their work also finallyputs to rest a problem that mathematicians had hoped would help settle the continuum hypothesis.Still, the overwhelming feeling among experts is that this apparently unresolvable proposition isfalse: While infinity is strange in many ways, it would be almost too strange if there weren’t manymore sizes of it than the ones we’ve already found.

Clarification: On September 12, this article was revised to clarify that mathematicians in the firsthalf of the 20th century wondered if the continuum hypothesis was true. As the article states, thequestion was largely put to rest with the work of Paul Cohen.

This article was reprinted on ScientificAmerican.com and Spektrum.de.

http://www.pnas.org/content/110/33/13238.full.pdf

http://www.pnas.org/content/110/33/13238.full.pdf

https://www.scientificamerican.com/article/mathematicians-measure-infinities-and-find-theyre-equal/

http://www.spektrum.de/news/von-unendlichkeit-zu-unendlichkeit/1507787

mathematicians measure infinities and find they’re equal · mathematicians subsequently used...

Documents