21世纪数学基础研究的最大进展之一
(2018-08-22 19:17:19)
21世纪数学基础研究的最大进展之一
袁萌
附:附:Mathematicians Measure Infinities and Find They’re Equal
Briefly, p is the minimum size of a collection of infinite sets of the natural numbers that have a “strong finite intersection property” and no “pseudointersection,” which means the subsets overlap each other in a particular way; t is called the “tower number” and is the minimum size of a collection of subsets of the natural numbers that is ordered in a way called “reverse almost inclusion” and has no pseudointersection.
The details of the two sizes don’t much matter. What’s more important is that mathematicians quickly figured out two things about the sizes of p and t. First, both sets are larger than the natural numbers. Second, p is always less than or equal to t. Therefore, if p is less than t, then p would be an intermediate infinity — something between the size of the natural numbers and the size of the real numbers. The continuum hypothesis would be false.
Mathematicians tended to assume that the relationship between p and t couldn’t be proved within the framework of set theory, but they couldn’t establish the independence of the problem either. The relationship between p and t remained in this undetermined state for decades. When Malliaris and Shelah found a way to solve it, it was only because they were looking for something else.
An Order of Complexity
Around the same time that Paul Cohen was forcing the continuum hypothesis beyond the reach of mathematics, a very different line of work was getting under way in the field of model theory.
H. Jerome Keisler invented “Keisler’s order.”
Courtesy of H. Jerome Keisler
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 the source code of mathematics. “I think the reason people are interested in classifying theories is they want to understand what is really causing certain things to happen in very different areas of mathematics,” said H. Jerome Keisler, emeritus professor of mathematics at the University of Wisconsin, Madison.
In 1967, Keisler introduced what’s now called Keisler’s order, which seeks to classify mathematical theories on the basis of their complexity. He proposed a technique for measuring complexity and managed to prove that mathematical theories can be sorted into at least two classes: those that are minimally complex and those that are maximally complex. “It was a small starting point, but my feeling 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 motivated in part by a desire to understand that question. Keisler describes complexity as the range of things that can happen in a theory — and theories where more things can happen are more complex than theories 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, little progress was made on Keisler’s order for 30 years.
Saharon Shelah is a co-author of the new proof.
Yael Shelah
Then, in her 2009 doctoral thesis and other early papers, Malliaris reopened the work on Keisler’s order and provided new evidence for its power as a classification program. In 2011, she and Shelah started working together to better understand the structure of the order. One of their goals was to identify more of the properties that make a theory maximally complex according to Keisler’s criterion.
Malliaris and
Shelah eyed two properties in particular. They already knew that
the first one causes maximal complexity. They wanted to know
whether the second one did as well. As their work progressed, they
realized that this question was parallel to the question of whether
p and t are equal. In 2016, Malliaris and Shelah published a
60-page paper that solved both problems: They proved that the two
properties are equally complex (they both cause maximal
complexity), and they proved 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 set theory. The honor reflects the surprising, and surprisingly powerful, nature of their proof. Most mathematicians had expected that p was less than t, and that a proof of that inequality would be impossible within the framework of set theory. Malliaris and Shelah proved that the two infinities are equal. Their work also revealed that the relationship between p and t has much more depth to it than mathematicians had realized.
“I think people thought that if by chance the two cardinals were provably equal, the proof would maybe be surprising, but it would be some short, clever argument that doesn’t involve building any real machinery,” said Justin Moore, a mathematician at Cornell University who has published a brief overview of Malliaris and Shelah’s proof.
Related:
To Settle Infinity Dispute, a New Law of Logic
Is Infinity Real?
Mathematicians Bridge Finite-Infinite Divide
Instead, Malliaris and Shelah proved that p and t are equal by cutting a path between model theory and set theory that is already opening new frontiers of research in both fields. Their work also finally puts 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 is false: While infinity is strange in many ways, it would be almost too strange if there weren’t many more sizes of it than the ones we’ve already found.
Clarification: On September 12, this article was revised to clarify that mathematicians in the first half of the 20th century wondered if the continuum hypothesis was true. As the article states, the question was largely put to rest with the work of Paul Cohen.
This article was reprinted on ScientificAmerican.com and Spektrum.de.(全文完)