无穷大研究中的一个新进展

sheng_jianguo

　

　　最近在网上看到外国人写的一篇关于无穷大研究的报道(国内好像无人报道)，觉得非常有意思。
　　为了让更多人共享此文，我试着翻译成中文后在这里发表。
　　由于本人的外语水平有限，故将原文附后。如发现翻译有不对地方一定要指出来，万分感谢！


无穷大
数学家度量两个无穷大后发现它们大小是相等的

INFINITY
Mathematicians Measure Infinities and Find They’re Equal

两位数学家已经证明两个不同无穷大的大小是相等的，解决了一个存在已久的问题。他们在建立于数学理论的复杂性和无穷大的大小之间有惊人的联系基础上完成了这个证明。

Two 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 the complexity of mathematical theories.

一个突破，推翻了几十年的传统观点。两数学家证明了两个不同元素的无穷大其实大小是一样的。这进展也触及了一个数学中最著名也是最棘手的问题：是否存在大小在自然数组成的无穷大和实数组成的无穷大之间的无穷大。

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

这个问题早在一个世纪以前就已经明确提出了。当时数学家们知道，“实数集比自然数集大，但不知大多少，它是比自然数集大的下一个集，还是另有一个集合大小在其中间？”芝加哥大学的马利亚里斯（Maryanthe Malliaris）和耶路撒冷希伯来大学及罗格斯大学的新研究合著者希拉（Saharon Shelah）说。

The problem was first identified over a century ago. At the time, mathematicians knew that “the real numbers 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 of the new work along with Saharon Shelah of the Hebrew University of Jerusalem and Rutgers University.

在他们的新研究中，马利亚里斯和希拉解决了涉及70年的老问题：是否一个无穷大（称之为p）小于另一个无穷大（称之为t）。他们证明了这两个无穷大的大小事实上是相等的，这使数学家们感到惊讶。

In their new work, Malliaris and Shelah resolve a related 70-year-old question about whether one infinity (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.

希拉说：“这当然是我的结论。一般的看法是，p应小于t。”

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

马利亚里斯和希拉的证明去年在美国数学学会杂志上发表，今年七月荣获集合论领域一个最高奖项。但是他们的工作所带来的影响远远超出了与这两个无穷大相关的具体问题。它为无限集合的大小和平行映射于数学理论的复杂性之间开辟了一条意外的联系通道。

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

许多无穷大
Many Infinities

无穷大的概念是令人费解的。然而有不同大小的无穷大的想法不同，这可能是有史以来最违反直觉的数学发现。事实上，在一个匹配的游戏中连小孩子都能理解。

The 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.

假设你有两组对象或者两个像数学家们所称的 “集合”：一个汽车集合和一个司机集合。如果每辆车都配有一个司机，没有空车，也没有司机遗留，那么你能确信汽车的数量等于司机的数量（即使你不知道具体数量是多少）。

Suppose you have two groups of objects, or two “sets,” as mathematicians would call them: a set of cars and a set of drivers. If there is exactly 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 you don’t know what that number is).

十九世纪末，德国数学家康托（Georg Cantor）在数学的形式语言中捕捉到了这种匹配策略的精髓。他证明了当它们的元素之间可以一一对应时，即每辆车都正好对应一个司机，这两集合具有相同的大小，或者说具有相同的“基数”。也许更令人惊讶的是，他证明这种处理方法同样适用于无限大的集合。

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

考虑自然数：1, 2, 3等等。自然数的集合是无限的。但是，关于偶数的集合，或者仅仅是质数呢？这些集合的每一个初看起来都是自然数的一小部分。事实上，在这些集合沿数轴的任何有限扩展中，偶数大约有自然数的一半而质数甚至更少。
1  2  3  4  5……（自然数）
2  4  6  8  10……（偶数）
2  3  5  7  11……（素数）

Consider the natural numbers: 1, 2, 3 and so on. The set of natural numbers is infinite. But what about the set of just the even numbers, or just the prime numbers? Each of these sets would at first seem to be a smaller subset of the natural numbers. And indeed, over any finite stretch of the number line, there are about half as many even numbers as natural numbers, and still fewer primes.
1       2       3       4       5       …      (natural numbers)
2       4       6       8       10     …      (evens)
2       3       5       7       11     …      (primes)

然而，无限集的表现不同。康托表明这些无限集中的元素之间有一一对应关系。

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

正因为如此，康托得出结论：这三组集的大小都是一样的。数学家称这种大小的集合是“可数的”，因为你可以在每个这样的集合中给每个元素分配一个计数。

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

通过集合间一一对应关系建立了无穷集合的大小后，康托完成了一个更大的创举：他证明了一些无限集甚至比自然数集还要大。

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

考虑数轴上的所有点组成的实数集。实数集有时被称为“连续统”，这反映了其具有连续性：一个实数与下一个实数之间没有间隙。康托能够证明实数集不能和自然数集一一对应：即使你创建了一个实数配对自然数的无限对应表，但总能找出另外一个实数不在你列的表中。因此，他断定实数集要比自然数集大。这样，诞生了第二种无穷大：不可数无穷大。

Consider the real numbers, which are all the points on the number line. The real numbers are sometimes referred to as the “continuum,” reflecting their continuous 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-one correspondence with the natural numbers: Even after you create an infinite list pairing natural numbers with real numbers, it’s always possible to come up with another real number that’s not on your list. Because of this, he concluded that the set of real numbers is larger than the set of natural numbers. Thus, a second kind of infinity was born: the uncountably infinite.

康托无法推出：是否存在一个中间大小的无穷大——大小介于可数自然数集和不可数实数集之间的集合。他猜测不存在，这就是现在被称为的连续统假设猜想。

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. He guessed not, a conjecture now known as the continuum hypothesis.

1900年德国数学家希尔伯特（David Hilbert）列出了数学中最重要的23个问题。他把连续统假设放在首位。马利亚里斯说：“这似乎是一个明显需要迫切解决的问题。”

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

这个世纪以来，这个问题已证明几乎是对数学家最大努力的独特阻碍。存在之间的无穷大吗？我们也许永远不会知道。

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

力迫法问世
Forced Out

整个上半二十世纪，数学家们都试图通过研究数学许多领域中出现的各种无穷集来解决连续统假设。他们希望可能通过比较这些无穷大，开始理解在自然数集的大小和实数集的大小之间存在实在间隔的可能性。

Throughout the first half of the 20th century, mathematicians tried to resolve the continuum hypothesis by studying various infinite sets that appeared in many areas of mathematics. They hoped that by comparing these infinities, they might start to understand the possibly non-empty space between the size of the natural numbers and the size of the real numbers.

许多比较被证明很难刻画出来。20世纪60年代，数学家科恩（Paul Cohen）解释了为什么会这样。科恩提出了一种称为“力迫”的方法，证明了连续统假设独立于数学公理——即连续统假设不能在集合论的框架内加以证明。（科恩的研究的互补工作是哥德尔（Kurt Gödel） 在20世纪40年代的研究，表明不能在数学通常的公理体系中证明连续统假设为假。）

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

科恩的研究为他赢得了1966年度的菲尔兹（Fields）数学奖（数学最高荣誉之一）。随后数学家们用力迫法解决了许多已经提出了过去半个多世纪的无穷大之间的比较问题，表明这些问题也不能在集合论框架（特别是策梅洛·弗伦克尔（Zermelo Fraenkel）集合论加上选择公理）内解答。

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

然而，有些问题依然存在，包括20世纪40年代提出的关于p是否等于t的问题。p和t是无穷大的序数，它们以清晰和似乎独特的方式量化自然数集中最小大小的一些子集集合。

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

（简而言之，p是一个具有“强有限交性”和没有“伪交性”的自然数无穷子集合组成的最小规模的集合，这意味着其中子集以一个特定的方式相互重叠；称为“塔数”的t是按“反向几乎包含”且没有“伪交性”的自然数无穷子集合组成的最小规模有序集合）

（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.）

这两个集的大小在细节上没有多大关系。更重要的是，数学家们很快就弄清楚了p和t的大小。第一，这两个集合都比自然数集大。第二，p总是小于或等于t。因此，如果p小于t，那么p将是一个介于自然数集和实数集大小之间的中间无穷大，说明连续统假设是假的。

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.

数学家们倾向于p和t之间关系的假设不能在集合论的框架内证明，但也不能确定p小于t或p等于t的独立性。几十年来，p和t的关系一直处于这种不确定状态中。当马利亚里斯和希拉发现了一种解决它的方法后，数学家们只能寻找其它东西来研究了。

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.

复杂的序数
An Order of Complexity

大约在科恩将连续统假设力迫法的范围扩展到数学中各领域的同一时期，一个非常不同方向的研究工作正在模型论领域中进行。

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.

对于一个模型论家来说，“理论”是数学领域定义的一组公理或规则。你可以把模型论看作是对数学理论进行分类的一种方法——对数学源码（表达式）的探索。麦迪逊市威斯康星大学的名誉数学教授基斯勒（H. Jerome Keisler）说：“我认为人们对分类理论感兴趣是他们想了解什么是真正造成某些事情在数学非常不同领域发生的原因。”

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.

1967年, 基斯勒引入了一种现在被称为基斯勒序数概念，其在复杂性基础上探索数学理论分类。他提出了一种度量复杂性的技术，并设法证明数学理论至少可以分为两类：最小复杂性和最大复杂性。基斯勒说：“这是一个小的起点，但在这一点上，我的感觉是会有无限多的类。”

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.

一个理论复杂性的含义并不总是显而易见的。这个领域的许多研究在某种程度上都是出于要理解这个问题的动机。基斯勒在理论中可能发生事情的范围中描述复杂性，理论中可能发生更多的事情要比理论中可能发生较少事情更复杂。

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.

基斯勒引入了他的序数十多年后，希拉发表了一本很有影响力的书。其中重要的一章显示了复杂性中从不太复杂理论到更复杂的理论的区分界限自然发生的跳跃。除此之外，这方面研究在基斯勒序数创建30年之后没有取得什么进展。

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.

然后，马利亚里斯在她的2009年博士论文和其他早期的论文中重新开始了基斯勒序数研究并用一个分类程序为基斯勒序数的幂提供新的证据。2011年，她和希拉为更好地了解序数的结构开始一起工作。他们的目标之一是确定更多的性质来建立一个按基斯勒准则最复杂的理论。

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.

马利亚里斯和希拉特别注重两个特性。他们已经知道第一个特性会导致最大的复杂性。他们想知道第二个特性是否也同样如此。随着工作进展，他们意识到这个问题与p和t是否相等的问题是平行关联的。2016年马利亚里斯和希拉发表一篇60页的论文，解决了这两个问题：他们证明了这两种性质具有同样复杂性（都造成极大的复杂性）及证明了p等于t。

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.

马利亚里斯说：“想方设法使一切东西排序，这是被解决问题的特征。”

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

今年七月，马利亚里斯和希拉获得了豪斯道夫（Hausdorff）奖，这是集合论中的最高奖项之一。这一荣誉反映了他们的证明令人十分惊讶。大多数数学家都期望p小于t，这个不等式在集合论的框架内是不可能被证明的。马利亚里斯和希拉证明了这两种无穷大的大小是相等的。他们的研究工作也揭示了p和t之间的关系比数学家们意识到的要深刻得多。

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.

一位发表了马利亚里斯和希拉证明的简要概述的康奈尔大学数学家摩尔（Justin Moore）说：“我认为人们会想是否有可能这两个基数被证明相等，但证明能会令人惊讶又精巧不太长且不涉及建立在任何实在的机器上。”

“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.

马利亚里斯和希拉的证明不仅是由模型论和集合论之间的联系得出p和t相等，而且已在这两个理论中开辟了新的研究领域。他们的研究也终结了数学家们希望解决的一个问题，并将有助于解决连续统假设。不过，专家们压倒性的直觉认为，几乎无法解决的连续统假设命题是错误的：无穷大在许多方面对我们来说都是陌生的，如果在已发现的无穷大之间没有更多大小不同的无穷大似乎是太奇怪了。

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.

说明：9月12日，这篇论文已修改，以阐明数学家们在上半二十世纪想知道的连续统假设是否正确。正如文中所述，这个问题在很大程度上取决于科恩的研究工作。

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.

本文是转载，选自 ScientificAmerican.com和 Spektrum.de。

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

附注：
本文译自网文“Mathematicians Measure Infinities and Find They’re Equal”网址：
https://www.quantamagazine.org/mathematicians-measure-infinities-find-theyre-equal-20170912/

两位数学家论文“COFINALITY SPECTRUM THEOREMS IN MODEL THEORY, SET THEORY AND GENERAL TOPOLOGY”下载地址：http://math.uchicago.edu/~mem/Malliaris-Shelah-CST-new.pdf

wayne

搂了一眼，好像是根据Model Theory里的关于复杂度的定义，推理出实数和自然数的个数是等势的

zeroieme

实数和自然数的个数是等势的

畅想一下，那么所有实数可以用有限长度的自然数向量表示，于是数值计算的“实数计算”就可以是真的实数计算不是若干有效数字。

wayne

zeroieme 发表于 2017-12-17 00:38
畅想一下，那么所有实数可以用有限长度的自然数向量表示，于是数值计算的“实数计算”就可以是真的实数计 ...

斯坦福大学的官网有关于Model Theory的介绍。https://plato.stanford.edu/entries/model-theory/
大致意思就是：Model Theory是以Alfred Tarski对真命题的定义作为理论基础[paradigm]，运用集合论的相关理论结构，来研究一般性的语言的理解问题，包括形式语言和自然语言。

model theory is the study of the interpretation of any language, formal or natural, by means of set-theoretic structures, with Alfred Tarski's truth definition as a paradigm.

sheng_jianguo

关于1#给出的“无穷大研究中的一个新进展”报道，个人有几点补充想法：
　　1. 文章是一位学数学的网友介绍的。他看后说：“他们说整数的集合与实数的集合是一样多的”（我们这里好像也有类似看法）。这是我绝对无法接受的论点。
　　学数学的都应该很清楚，在实无穷和“一一对应”定义的无穷集合大小观点下，自然数（或整数）集小于实数集的证明是非常严谨的。（可以说证明是天衣无缝的。有些人自称证明了自然数集和实数集等势，结果不难发现所谓的“证明”都有明显的错误）
　　1#报道和两位数学家论文并没有说：整数的集合与实数的集合是一样多的。相反给出了整数的集合与实数的集合不是一样多的Cantor严谨证明方法。P和t对应的基数都大于自然数集对应的基数且不超过实数集对应的基数，即Z＜p,t≤R，且p≤t。如果能严格证明p＜t（数学家们以前都是认为p＜t是真的），则就可得出连续统假设（世上最著名的数学问题）是假的，这将是全世界数学史上最最伟大的发现，估计为此可能世上数学界至少要放假庆祝一星期。可惜的是这两位数学家证明了p＝t，不严格说，这只是给连续统假设成立增加了一点点概率，并不能说明连续统假设是真或假，更不能由此说明自然数的集合与实数的集合是一样多的。
　　2. 报道原文中还有两张照片，不知为什么在1#总是传不上去。这里再试传一下：

引入基斯勒序数的基斯勒（H. Jerome Keisler）

新证明的合著者希拉（Saharon Shelah）

　　我感觉奇怪的是：这两位的照片都有了，为什么没有放报道的主角——马利亚里斯的照片呢？如果有马利亚里斯的照片，文章就更完美了。

mathe

前面描述自然数和实数等势显然是有问题的。但是根据后面对p,t的定义，显然是不同的东西。这个问题的确同连续统假设相关。但是连续统假设不是已经被证明无法证明也无法否定了吗？

zeroieme

mathe 发表于 2017-12-17 16:34
前面描述自然数和实数等势显然是有问题的。但是根据后面对p,t的定义，显然是不同的东西。这个问题的确同连 ...

连续统假设已经被证明在现有公理系统下无法证明也无法否定。如果引入一条能让多数数学家认可的新公理，又能对连续统假设作断言。就是重大进展。

sheng_jianguo

mathe 发表于 2017-12-17 16:34
前面描述自然数和实数等势显然是有问题的。但是根据后面对p,t的定义，显然是不同的东西。这个问题的确同连 ...

前面描述的一段容易误解为自然数和实数等势问题，实际上描述的是p和t等势问题。
关于连续统假设的确是一个很有意思问题（有机会可在有关专题中专门论述），这里简单说一下我的观点：
数学中命题（或论断）一般可分为两类
1. 不确定型
典型的命题如：三角形内角之和等于180°。在欧氏几何中是真的，而在非欧氏几何中是假的，我认为这不是数学的不确定性，而是给出这个命题时，一些基本定义没在命题中说清楚，比如：两点间的直线的定义。如果命题中所有的概念都是确定的，那么此命题也就转变成确定型了。
2. 确定型
典型的命题如：哥德巴赫猜想。虽然我们现在还不知道其是真是假，但其是真是假两者之间只有一个正确的且必定有一个是正确的。也就是哥德巴赫猜想是真是假是确定的（虽然由于人类的一些条件受限而现在不能确定其真假）。如果在增加某个公理下哥德巴赫猜想为真，而在增加另一个公理下哥德巴赫猜想为假，这不能说明哥德巴赫猜想不确定性，反过来说明这两个增加的公理中，至少有一个是错的。
回到连续统假设，我认为连续统假设是确定型命题，即连续统假设是真是假完全是确定的（就如哥德巴赫猜想）。因为所有关于连续统假设中定义都是确定的。也有学数学的没仔细分析错误认为连续统假设是不确定型命题，关于这一点有机会再详细论述。
连续统假设的最好结果大家都知道：在现有公理系统（ZFC）下，无法证明它是真的，也无法证明它是假的。但这决不能推出连续统假设是不确定型命题，只能说明采用现有的一些手段，是无法得出其是真是假。
上面已说过，对于像连续统假设这样的确定型命题，采用ZFC中增加公理的方法来解决问题不是好办法（而且也不一定能彻底解决）。
如果连续统假设是假的（大多数数学家都直觉相信连续统假设是假的），由于人类的智慧是无法估计的，找到一个具体集合X，其势满足|Z|<|X|<|R|是有可能的，这是一个彻底解决连续统假设的好方法，但难度非常非常大。有志于攻克连续统假设者，这也许是一条解决问题的路径。

zeroieme

保罗·柯恩以力迫法证明了，在现有公理系统（ZFC）下，无法证明它是真的，也无法证明它是假的。那么不增加公理而找到具体集合X，意味着推翻了保罗·柯恩的证明。

sheng_jianguo

zeroieme 发表于 2017-12-17 22:04
保罗·柯恩以力迫法证明了，在现有公理系统（ZFC）下，无法证明它是真的，也无法证明它是假的。那么不增加 ...

严谨的说法如下
柯恩在1963年证明：如果公理系统ZFC是协调的，则在ZFC下推不出连续统假设是真的。
之前，哥德尔已证明：如果公理系统ZFC是协调的，则在ZFC下推不出连续统假设是假的。
虽然ZFC系统能得出数学（特别是集合论）中大量正确命题，但不是说已有的正确数学命题都可以在ZFC系统下得出。
ZFC系统有自己的符号、公式（包括公理）和有限条推理规则。在ZFC下推出命题A为真是指在ZFC系统中按推理规则在有限步内得出A为真。所以说ZFC系统下推出命题A为真要求是很高的（为保证ZFC系统下推出命题A为真在任何现实世界中不会为假），不能说ZFC系统下推不出命题B是真是假就不可能用其它正确的数学方法（事实上证明是可靠的）来证明B是真是假。也就是说如果能用其它正确方法证明连续统假设为假，不能由此得出推翻了柯恩的证明。