Why Math’s Final Axiom Proved So Controversial

Kristina Armitage/Quanta Magazine

克里斯蒂娜·阿米蒂奇/广达杂志

H ow do mathematicians decide that something is true?

数学家如何确定某件事是真的？

They write a proof.

他们写了一个证明。

Often they start with proofs that already exist, building on or drawing connections between proven claims.

他们通常从已经存在的证据开始，建立在已证实的主张之间或在已证实的主张之间建立联系。

Each of these proofs, in turn, has relied on other proofs to make its point, and so on.

反过来，这些证据中的每一个都依赖于其他证据来证明其观点，等等。

Proofs upon proofs.

证据之上的证据。

Truths upon truths.

真理之上的真理。

But eventually this process must come to an end.

但最终这个过程必须结束。

At some point, things are true simply because they are.

在某些时候，事情就是真的，因为它们就是真的。

These truths are the axioms, the ground rules.

这些真理是公理、基本规则。

And it is tempting to stop there — to declare, as Penelope Maddy , a philosopher of mathematics at the University of California, Irvine, put it, “that axioms are obvious or intuitive or conceptual truths.”

人们很容易就此止步——正如加州大学欧文分校数学哲学家佩内洛普·麦迪所说，“公理是显而易见的、直观的或概念性的真理。”

After all, most mathematicians simply accept that their work relies on an axiomatic system — namely, “Zermelo-Fraenkel set theory with the axiom of choice,” or ZFC — if they bother to acknowledge the axioms at all.

毕竟，大多数数学家只要愿意承认这些公理，就简单地接受他们的工作依赖于公理系统，即“具有选择公理的策梅洛-弗兰克尔集合理论”或 ZFC。

ZFC is a list of 10 basic principles that together form the foundation on which nearly all of modern mathematics is built.

ZFC 列出了 10 条基本原理，它们共同构成了几乎所有现代数学的基础。

But a closer inspection reveals a more unsettled, human process of establishing truth.

但仔细观察就会发现，建立真相的过程更加不稳定、更人性化。

“Any honest, clear-eyed examination of how the axioms of ZFC came to be adopted would have to acknowledge that a wide range of mathematical considerations went into these decisions,” Maddy said.

“任何对 ZFC 公理如何被采用的诚实、清晰的研究都必须承认这些决定涉及广泛的数学考虑，”Maddy 说。

That process, which began over a century ago, is still very much in progress.

这一进程始于一个多世纪前，目前仍在进行中。

The late 1800s were a time of paradoxes and doubt, the result of mathematicians beginning to search for cohesive ideas about what rules the mathematical universe obeyed.

1800 年代末是一个充满悖论和怀疑的时代，这是数学家开始寻找关于数学宇宙遵循的规则的有凝聚力的想法的结果。

There were axiomatic systems out there, but they tended to be for specific areas of math: Euclid’s postulates for geometry; various schemes for standardizing arithmetic.

那里有公理系统，但它们往往适用于数学的特定领域：欧几里得的几何公设；标准化算术的各种方案。

But how did they all fit together?

但它们是如何组合在一起的呢？

Could all of math be derived from one common set of rules?

所有的数学都可以从一组通用的规则中推导出来吗？

Mathematicians found a potential solution — and more doubts — in the work of Georg Cantor.

数学家们在乔治·康托尔的研究中找到了一个潜在的解决方案，同时也找到了更多的疑问。

At the time, Cantor was studying the real numbers — that is, all the numbers that appear on the number line — and what they could say about the nature of infinity.

当时，康托正在研究实数——即数轴上出现的所有数字——以及它们对无穷大本质的描述。

He had found that there were more real numbers than whole numbers , giving rise to the profound realization that not all infinities are the same size.

他发现实数比整数多，从而深刻地认识到并非所有无穷大都具有相同的大小。

The German mathematician Ernst Zermelo came up with the axiom of choice, the most controversial of the 10 basic principles that today underlie all of mathematics.

德国数学家恩斯特·泽梅洛提出了选择公理，这是当今所有数学的 10 条基本原理中最具争议的一个。

To make this comparison, Cantor had used a seemingly simple tool: the set.

为了进行这种比较，康托使用了一个看似简单的工具：集合。

A set is a collection of objects, or elements.

集合是对象或元素的集合。

It might be a collection of numbers, like the real numbers, or a collection of shapes, or even a collection of other sets.

它可能是数字的集合，例如实数，也可能是形状的集合，甚至是其他集合的集合。

Over time, it became clear that complex and disparate mathematical ideas — nearly all of them — could be represented with these same elementary entities.

随着时间的推移，人们发现复杂且不同的数学思想——几乎所有的数学思想——都可以用这些相同的基本实体来表示。

As a result, the set emerged as a potential tool for ironing out any inconsistencies between different areas of math.

因此，该集合成为消除不同数学领域之间任何不一致的潜在工具。

But early set theory lacked canonical rules.

但早期的集合论缺乏规范规则。

It was possible to define sets with any property, which led to exactly the sorts of paradoxes that were bothering mathematicians at the time.

可以用任何属性来定义集合，这恰恰导致了当时困扰数学家的各种悖论。

Consider, for example, the set of all sets that are not members of themselves.

例如，考虑所有不属于其自身的集合的集合。

Does this set contain itself?

这个集合包含它自己吗？

Whether you answer yes or no, you get a contradiction now known as Russell’s paradox.

无论你的回答是或否，你都会遇到一个矛盾，现在被称为罗素悖论。

As mathematicians obsessed over these dilemmas, ZFC emerged out of a struggle with another idea of Cantor’s.

当数学家们痴迷于这些困境时，ZFC 在与康托尔的另一个想法的斗争中脱颖而出。

In 1883, Cantor introduced his “well-ordering principle.” He claimed that it should be possible to arrange any set so that all of its (non-empty) subsets would have a smallest element.

1883 年，康托提出了他的“良序原则”。他声称应该可以安排任何集合，使其所有（非空）子集都具有最小元素。

For finite sets, this is intuitive.

对于有限集，这是直观的。

You can always put the lesser items first.

您始终可以将次要的物品放在第一位。

But for infinite sets, it is less obvious.

但对于无限集，这一点就不那么明显了。

Take the set of integers {…, −2, −1, 0, 1, 2, …}.

取整数集 {…, −2, −1, 0, 1, 2, …}。

The negative numbers form a subset, but they also get lower and lower for eternity.

负数形成一个子集，但它们也会永远变得越来越低。

It seems as if there can be no least element.

似乎不可能有最小的元素。

But what if you arrange the original set of integers like this: {0, −1, 1, −2, 2, …}?

但是，如果您将原始整数集像这样排列：{0, −1, 1, −2, 2, …} 会怎样？

Now you can say that the smallest element is the one that comes first in any subset.

现在您可以说最小的元素是任何子集中第一个出现的元素。

In this way, −1 becomes the smallest element of the subset of negative numbers.

这样，-1就成为负数子集的最小元素。

Cantor’s “law” was that this should be possible for all sets, even if you can’t explicitly construct the proper ordering.

康托尔的“定律”是，这对于所有集合都应该是可能的，即使你无法明确地构建正确的排序。

It was one way of arguing that infinite sets behave like finite ones.

这是论证无限集的行为类似于有限集的一种方式。

In 1904, the German mathematician Ernst Zermelo proved it.

1904年，德国数学家恩斯特·策梅洛证明了这一点。

He did so by showing that Cantor’s law was equivalent to a principle he had developed while exploring the properties of sets.

他通过证明康托定律等同于他在探索集合性质时发展出的原理来做到这一点。

This principle, the so-called axiom of choice, said that if you start with multiple (or even infinitely many) non-empty sets, you can choose one element from each of those sets to create a new set.

这一原则，即所谓的选择公理，表示如果您从多个（甚至无限多个）非空集合开始，您可以从每个集合中选择一个元素来创建一个新集合。

Zermelo developed his other axioms to prove this equivalence.

策梅洛发展了他的其他公理来证明这种等价性。

“He was just listing all the assumptions that he needed to get the proof through,” said Joan Bagaria , a set theorist at the University of Barcelona.

“他只是列出了证明所需的所有假设，”巴塞罗那大学的集合论学家琼·巴加里亚 (Joan Bagaria) 说。

That list included the basic idea that there is such a thing as a set, which is defined by its elements.

该列表包含了这样的基本思想：存在集合这样的东西，它是由其元素定义的。

Other axioms dealt with the formation of sets from other sets, or with the existence of infinite sets.

其他公理涉及从其他集合形成集合，或涉及无限集合的存在。

Zermelo’s list of axioms emerged at a time when many mathematicians, such as Abraham Fraenkel, were also tinkering with set theory’s foundations.

策梅洛的公理列表出现之际，许多数学家（例如亚伯拉罕·弗兰克尔）也在修补集合论的基础。

Many of them found themselves arriving at different formulations of similar ideas — and some new ones, too, that resolved problems arising from newer theories having to do with larger forms of infinity.

他们中的许多人发现自己对类似的想法得出了不同的表述——还有一些新的表述，解决了与更大形式的无穷大有关的新理论所产生的问题。

In 1930, Zermelo released a “final” list that included revisions to his own axioms as well as a handful of additions — but not, at first, the axiom of choice.

1930 年，策梅洛发布了一份“最终”列表，其中包括对他自己的公理的修订以及一些补充，但一开始并不是选择的公理。

Mathematicians were more hesitant to include it, because unlike the other axioms, it defined sets without giving an explicit way to construct them.

数学家对于是否包含它更加犹豫，因为与其他公理不同，它定义了集合，但没有给出构造集合的明确方法。

Abraham Fraenkel was one of the founders of modern set theory, adding several key rules to its core list of axioms to help get rid of various paradoxes.

亚伯拉罕·弗兰克尔 (Abraham Fraenkel) 是现代集合论的创始人之一，他在其核心公理列表中添加了几条关键规则，以帮助摆脱各种悖论。

The David B.

大卫·B.

Keidan Collection of Digital Images from the Central Zionist Archives

Keidan 中央犹太复国主义档案馆收集的数字图像

Zermelo was pleased that his list of principles, known as ZF, appeared to cleanse the set-theoretic universe of many major paradoxes such as Russell’s.

策梅洛很高兴他的原则清单（称为 ZF）似乎清除了集合论宇宙中的许多主要悖论，例如罗素的悖论。

But he lamented that he was unable to prove that his axiomatic system was “consistent” — that it didn’t yield contradictions.

但他遗憾的是，他无法证明他的公理体系是“一致的”——它不会产生矛盾。

He need not have worried.

他不必担心。

Just a few years after the arrival of ZF , Kurt Gödel showed that no axiomatic system capable of basic arithmetic can be used to prove its own consistency.

ZF 到来后仅仅几年，库尔特·哥德尔就表明，没有任何能够进行基本算术的公理系统可以用来证明其自身的一致性。

Moreover, any consistent system must also be incomplete, meaning that there are true mathematical statements that cannot be proved using the system’s axioms.

此外，任何一致的系统也必定是不完备的，这意味着存在无法使用系统公理来证明的真实数学陈述。

In fact, in the 1960s, the Stanford mathematician Paul Cohen proved that the axiom of choice is “independent” of the other axioms — that is, under the rules of ZF, the axiom of choice cannot be proved true or false.

事实上，在 20 世纪 60 年代，斯坦福大学数学家 Paul Cohen 就证明了选择公理是“独立于”其他公理的——也就是说，在 ZF 规则下，选择公理无法被证明为真或假。

Once it was clear that logic could not validate the axiom of choice one way or the other, the question became: Is it useful?

一旦清楚逻辑无法以某种方式验证选择公理，问题就变成了：它有用吗？

And it was.

确实如此。

It makes a great deal of other math possible — especially math related to infinite objects .

它使许多其他数学成为可能——尤其是与无限对象相关的数学。

After that, the axiom of choice gained much more widespread acceptance.

此后，选择公理获得了更广泛的接受。

“Without choice, your tools are very limited,” Bagaria said.

“如果没有选择，你的工具就非常有限，”巴加里亚说。

“It’s like doing math with your hands tied behind your back.” And so the C (for “choice”) came to be affixed to the list of axioms that were originally developed to support it.

“这就像双手被绑在背后做数学一样。”因此，C（代表“选择”）被添加到最初为支持它而开发的公理列表中。

The axiom of choice demonstrates the folly of believing that mathematical axioms are self-evident or obvious.

选择公理表明，相信数学公理是不言而喻的或显而易见的是愚蠢的。

An axiom can also be accepted for plenty of other reasons, as Maddy put it — such as for its power to generate interesting theorems.

正如麦迪所说，公理还可以因为许多其他原因而被接受，例如它能够产生有趣的定理。

The ZFC axioms are often regarded as perhaps the most universal truths that humanity has managed to articulate — for while it may be possible for physicists to imagine universes in which physical laws are turned inside out, mathematical laws will remain constant.

ZFC 公理通常被认为可能是人类已经设法阐明的最普遍的真理——因为虽然物理学家可能会想象物理定律被彻底颠倒的宇宙，但数学定律将保持不变。

It is a paradox without resolution: The foundations of mathematics are as universal, as solid as anything humanity knows, a core part of nearly every mathematical truth.

这是一个没有解决方案的悖论：数学的基础与人类所知的一切一样普遍、坚实，是几乎所有数学真理的核心部分。

And yet they remain simply what we choose to believe.

然而它们仍然只是我们选择相信的东西。

Get highlights of the most important news delivered to your email inbox

获取发送到您的电子邮件收件箱的最重要新闻的摘要

Quanta Magazine moderates comments to facilitate an informed, substantive, civil conversation.

《Quanta》杂志对评论进行审核，以促进知情、实质性、文明的对话。

Abusive, profane, self-promotional, misleading, incoherent or off-topic comments will be rejected.

辱骂、亵渎、自我推销、误导、语无伦次或偏离主题的评论将被拒绝。

Moderators are staffed during regular business hours (New York time) and can only accept comments written in English.

版主在正常工作时间（纽约时间）有工作人员，并且只能接受英文评论。