去年，国际数学界发生一件大事，对此，国内无人报道，即康托尔连续统假设（CH）不成立。也就说，自然数与实数一样多。为此，数学教科书需要改写了。

1900年，希尔伯特提出23个数学难题，康托尔假设摆在第一的位置。直到上世纪60年代，Cohen利用“forcing”法，证明CH独立于ZFC公理系统，即不能证明其为真，也不能证明其为假。

1967年，基于模型论的研究，Keisler提出数学理论的“复杂度”理念，企图把数学理论按照其复杂度分类。

2006年，30年过去了，有人重新翻开Keisker复杂度理论，发现它与康托尔假设有关，2016年，发表了600页的长篇论文，并于2017年荣获豪斯多夫大奖。

请见科普文章“MathematiciansMeasure Infinities and Find They’re Equal（此文发表于2017年9月12日）”

老实说，数学就是研究无穷的的学问。百度文库发表的相关文章，全是胡说八道！

袁萌  7月1日

infinity Mathematicians Measure Infinities and Find They’re Equal

By Kevin Hartnett

September 12, 2017

Twomathematicians have proved that two different infinities are equal in size,settling a long-standing question. Their proof rests on a surprising linkbetween the sizes of infinities and the

complexity ofmathematical theories.

Colors Collective for Quanta Magazine

In a breakthrough that disproves decades of conventional wisdom, twomathematicians have shown that two different variants of infinity are actuallythe same size. The advance touches on one of the most famous and intractableproblems in mathematics: whether there exist infinities between the infinitesize of the natural numbers and the larger infinite size of the real numbers.

The problem wasfirst identified over a century ago. At the time, mathematicians knew that “thereal numbers are bigger than the natural numbers, but not how much bigger. Isit the next biggest size, or is there a size in between?” said

Maryanthe Malliaris

of the University of Chicago, co-author of the new work along with

Saharon Shelah

of the Hebrew University of Jerusalem and Rutgers University.

In their new work, Malliaris and Shelah resolve a related 70-year-oldquestion about whether one infinity (call it p) is smaller than anotherinfinity (call it t). They proved the two are in fact equal, much to the surpriseof 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 ofthe American Mathematical Society and were honored this past July with one ofthe top prizes in the field of set theory. But their work has ramifications farbeyond the specific question of how those two infinities are related. It opensan unexpected link between the sizes of infinite sets and a parallel effort tomap the complexity of mathematical theories.

Many Infinities

The notion of infinity is mind- bending. But the idea that there can bedifferent sizes of infinity? That’s perhaps the most counterintuitivemathematical discovery ever made. It emerges, however, from a matching gameeven kids could understand.

Suppose you have two groups of objects, or two “sets,” asmathematicians would call them: a set of cars and a set of drivers. If there isexactly one driver for each car, with no empty cars and no drivers left behind,then you know that the number of cars equals the number of drivers (even if youdon’t know what that number is).

In the late 19th century, the German mathematician Georg Cantorcaptured the spirit of this matching strategy in the formal language ofmathematics. He proved that two sets have the same size, or “cardinality,” whenthey can be put into one-to-one correspondence with each other — when there isexactly one driver for every car. Perhaps more surprisingly, he showed thatthis approach works for infinitely large sets as well.

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

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

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

After heestablished that the sizes of infinite sets can be compared by putting theminto one-to-one correspondence with each other, Cantor made an even biggerleap: He proved that some infinite sets are even larger than the set of naturalnumbers.

Lucy Reading-Ikkanda/Quanta Magazine

Consider the real numbers, which are all the points on the number line.The real numbers are sometimes referred to as the “continuum,” reflecting theircontinuous nature: There’s no space between one real number and the next.Cantor was able to show that the real numbers can’t be put into a one-to-onecorrespondence with the natural numbers: Even after you create an infinite listpairing natural numbers with real numbers, it’s always possible to come up withanother real number that’s not on your list. Because of this, he concluded thatthe set of real numbers is larger than the set of natural numbers. Thus, asecond kind of infinity was born: the uncountably infinite.

What Cantor couldn’t figure out was whether there exists anintermediate size of infinity — something between the size of the countablenatural numbers and the uncountable real numbers. He guessed not, a conjecturenow known as the continuum hypothesis.

In 1900, the German mathematician David Hilbert made a list of 23 ofthe most important problems in mathematics. He put the continuum hypothesis atthe top. “It seemed like an obviously urgent question to answer,” Malliarissaid.

In the century since, the question has proved itself to be almostuniquely resistant to mathematicians’ best efforts. Do in-between infinitiesexist? We may never know.

Forced Out

Throughout thefirst half of the 20th century, mathematicians tried to resolve the continuumhypothesis by studying various infinite sets that appeared in many areas ofmathematics. They hoped that by comparing these infinities, they might start tounderstand the possibly non-empty space between the size of the natural numbersand the size of the real numbers.

Many of the comparisons proved to be hard to draw. In the 1960s, themathematician Paul Cohen explained why. Cohen developed a method called“forcing” that demonstrated that the continuum hypothesis is independent of theaxioms of mathematics — that is, it couldn’t be proved within the framework ofset theory. (Cohen’s work complemented work by Kurt Gödel in 1940 that showedthat the continuum hypothesis couldn’t be disproved within the usual axioms ofmathematics.)

Cohen’s work won him the Fields Medal (one of math’s highest honors) in1966. Mathematicians subsequently used forcing to resolve many of thecomparisons between infinities that had been posed over the previoushalf-century, showing that these too could not be answered within the frameworkof set theory. (Specifically, Zermelo-Fraenkel set theory plus the axiom ofchoice.)

Some problems remained, though, including a question from the 1940sabout whether p is equal to t. Both p and t are orders of infinity thatquantify the minimum size of collections of subsets of the natural numbers inprecise (and seemingly unique) ways.

Briefly, p is the minimum size of a collection of infinite sets of thenatural numbers that have a “strong finite intersection property” and no“pseudointersection,” which means the subsets overlap each other in aparticular way; t is called the “tower number” and is the minimum size of acollection 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 importantis that mathematicians quickly figured out two things about the sizes of p andt. First, both sets are larger than the natural numbers. Second, p is alwaysless than or equal to t. Therefore, if p is less than t, then p would be anintermediate infinity — something between the size of the natural numbers andthe size of the real numbers. The continuum hypothesis would be false.

Mathematicians tended to assume that the relationship between p and tcouldn’t be proved within the framework of set theory, but they couldn’t establishthe independence of the problem either. The relationship between p and tremained in this undetermined state for decades. When Malliaris and Shelahfound a way to solve it, it was only because they were looking for somethingelse.

An Order of Complexity

Around the sametime that Paul Cohen was forcing the continuum hypothesis beyond the reach ofmathematics, a very different line of work was getting under way in the fieldof model theory.

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

Courtesy of H. JeromeKeisler

For a model theorist, a “theory” is the set of axioms, or rules, thatdefine an area of mathematics. You can think of model theory as a way toclassify mathematical theories — an exploration of the source code ofmathematics. “I think the reason people are interested in classifying theoriesis they want to understand what is really causing certain things to happen invery different areas of mathematics,” said H. Jerome Keisler, emeritusprofessor of mathematics at the University of Wisconsin, Madison.

In 1967, Keislerintroduced what’s now called Keisler’s order, which seeks to classifymathematical theories on the basis of their complexity. He proposed a techniquefor measuring complexity and managed to prove that mathematical theories can besorted into at least two classes: those that are minimally complex and thosethat are maximally complex. “It was a small starting point, but my feeling atthat point was there would be infinitely many classes,” Keisler said.

It isn’t always obvious what it means for a theory to be complex. Muchwork 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 theorieswhere fewer things can happen.

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

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

Yael Shelah

Then, in her 2009doctoral thesis and other early papers, Malliaris reopened the work onKeisler’s order and provided new evidence for its power as a classificationprogram. In 2011, she and Shelah started working together to better understandthe structure of the order. One of their goals was to identify more of theproperties that make a theory maximally complex according to Keisler’scriterion.

Malliaris and Shelah eyed two properties in particular. They alreadyknew that the first one causes maximal complexity. They wanted to know whetherthe second one did as well. As their work progressed, they realized that thisquestion 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: Theyproved that the two properties are equally complex (they both cause maximalcomplexity), and they proved that p equals t.

“Somehow everything linedup,” 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 inset 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 ofset theory. Malliaris and Shelah proved that the two infinities are equal.Their work also revealed that the relationship between p and t has much moredepth to it than mathematicians had realized.

“I think people thought thatif by chance the two cardinals were provably equal, the proof would maybe besurprising, but it would be some short, clever argument that doesn’t involvebuilding any real machinery,” said Justin Moore, a mathematician at CornellUniversity 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 cuttinga path between model theory and set theory that is already opening newfrontiers of research in both fields. Their work also finally puts to rest aproblem that mathematicians had hoped would help settle the continuumhypothesis. Still, the overwhelming feeling among experts is that thisapparently unresolvable proposition is false: While infinity is strange in manyways, it would be almost too strange if there weren’t many more sizes of itthan the ones we’ve already found.

Clarification: On September 12, this article was revised to clarifythat mathematicians in the first half of the 20th century wondered if thecontinuum hypothesis was true. As the article states, the question was largelyput to rest with the work of Paul Cohen.

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

Kevin Hartnett

Senior Writer

September 12, 2017

continuum hypothesis

foundations of mathematics

infinity

mathematical logic

mathematics

model theory

set theory

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去年，国际数学界发生一件大事，对此，国内无人报道，即康托尔连续统假设（CH）不成立。也就说，自然数与实数一样多。为此，数学教科书需要改写了。                1900年，希尔伯特提出23个数学难题，康托尔假设摆在第一的位置。直到上世纪60年代，Cohen利用“forcing”法，证明CH独立于ZFC公理系统，即不能证明其为真，也不能证明其为假。... (A+B)%M = (A%M+B%M) % M (A*B)%M = (A%M*B%M) % M (A/B)%M = (A*C)%M =(A%M*C%M) % M，其中 B*C≡1(mod M)，B、M互质，C称为 B的逆元 (A/B)%M 的推导：(A/B)%M = (A/B) * 1 % M = (A/B)*B*C % M = (A*C) % M 【威尔逊定理】