What Do Gödel’s Incompleteness Theorems Truly Mean?

Kurt Gödel Papers, the Shelby White and Leon Levy Achives Center, Institute for Advanced Study; Samuel Velasco and Michael Kanyongolo/ Quanta Magazine

库尔特·哥德尔论文，谢尔比·怀特和莱昂·利维档案中心，高等研究院； Samuel Velasco 和 Michael Kanyongolo/Quanta 杂志

I n 1931, by turning logic on itself, Kurt Gödel proved a pair of theorems that transformed the landscape of knowledge and truth.

1931 年，库尔特·哥德尔将逻辑转向自身，证明了一对定理，改变了知识和真理的面貌。

These “incompleteness theorems” established that no formal system of mathematics — no finite set of rules, or axioms, from which everything is supposed to follow — can ever be complete.

这些“不完备性定理”表明，没有任何正式的数学系统——没有一组有限的规则或公理，一切都应该遵循这些规则或公理——可以是完整的。

There will always be true mathematical statements that don’t logically follow from those axioms.

总会有一些真正的数学陈述在逻辑上不符合这些公理。

I spent the early weeks of the Covid pandemic learning how the 25-year-old Austrian logician and mathematician did such a thing, and then writing a rundown of his proof in fewer than 2,000 words.

在新冠病毒大流行的最初几周里，我学习了这位 25 岁的奥地利逻辑学家和数学家是如何做到这一点的，然后用不到 2000 字写下了他的证明概要。

(My wife, when I reminded her of this period: “Oh yeah, that time you almost went crazy?” A slight exaggeration.)

（我的妻子，当我提醒她这段时期时：“哦，是的，那一次你几乎要疯了？”有点夸张。）

I n philosophy, “qualia” refers to the subjective qualities of our experience: what it’s like for Alice to see blue or for Bob to feel delighted.

在哲学中，“感受性”指的是我们体验的主观品质：爱丽丝看到蓝色或鲍勃感到高兴是什么感觉。

Qualia are “the ways things seem to us,” as the late philosopher Daniel Dennett put it.

正如已故哲学家丹尼尔·丹尼特所说，感受性是“事物在我们看来的方式”。

In these essays, our columnists follow their curiosity, and explore important but not necessarily answerable scientific questions.

在这些文章中，我们的专栏作家追随他们的好奇心，探索重要但不一定可以回答的科学问题。

But even after grasping the steps of Gödel’s proof, I was unsure what to make of his theorems, which are commonly understood as ruling out the possibility of a mathematical “theory of everything.” It’s not just me.

但即使在掌握了哥德尔证明的步骤之后，我仍然不确定如何理解他的定理，这些定理通常被理解为排除数学“万有理论”的可能性。不只是我。

In Gödel’s Proof (a classic 1958 book that I heavily relied upon for my account), philosopher Ernest Nagel and mathematician James R.

在《哥德尔证明》（一本 1958 年出版的经典著作，我的论述中非常依赖它）中，哲学家欧内斯特·内格尔 (Ernest Nagel) 和数学家詹姆斯·R.

Newman wrote that the meaning of Gödel’s theorems “has not been fully fathomed.”

纽曼写道，哥德尔定理的含义“尚未被完全理解”。

Maybe not, but six decades have passed since then.

也许不是，但从那时起已经过去了六年。

Where are we with these ideas today?

今天我们对这些想法的看法如何？

Recently, I asked logicians, mathematicians, philosophers, and one physicist to discuss the meaning of incompleteness.

最近，我请逻辑学家、数学家、哲学家和一位物理学家讨论不完备性的含义。

They had plenty to say about the implications of Gödel’s strange intellectual achievement and how it changed the course of humanity’s unending search for truth.

他们对哥德尔奇怪的智力成就的影响以及它如何改变人类永无止境地寻求真理的进程有很多话要说。

PANU RAATIKAINEN , philosopher at Tampere University and author of the Stanford Encyclopedia of Philosophy entry on Gödel’s incompleteness theorems

PANU RAATIKAINEN，坦佩雷大学哲学家，斯坦福哲学百科全书哥德尔不完备性定理条目的作者

Ever since the ancient Greeks, the axiomatic method has been widely taken as the ideal way of organizing scientific knowledge.

自古希腊以来，公理化方法就被广泛视为组织科学知识的理想方式。

The aim is to have a small number of “self-evident” basic propositions — axioms, principles, or laws — such that all truths of the field in question can be logically derived from them.

其目的是拥有少量“不言而喻”的基本命题——公理、原理或定律——以便可以从它们逻辑上推导出相关领域的所有真理。

Gödel’s incompleteness theorems show with mathematical precision that this ideal necessarily fails for large parts of mathematics.

哥德尔不完备定理以数学精度表明，这种理想对于大部分数学来说必然是失败的。

The whole of mathematical truth concerning even just positive integers (1, 2, 3 …) is so perplexingly complex that it does not follow from any finite set of axioms.

即使只是关于正整数（1、2、3……）的整个数学真理也非常复杂，以至于它不能从任何有限的公理集得出。

This means that some mathematical problems are not even in principle solvable by our current mathematical methods.

这意味着有些数学问题甚至原则上无法用我们当前的数学方法解决。

Progress may require creative conceptual innovation.

进步可能需要创造性的概念创新。

As a result, mathematical truths do not make up a unified whole of equally indubitable truths; instead, their status as knowledge varies gradually from doubtless facts to increasingly uncertain hypotheses.

因此，数学真理并不是由同样不容置疑的真理组成的统一整体。相反，它们作为知识的地位逐渐从毫无疑问的事实转变为越来越不确定的假设。

Raatikainen makes a good point that Gödel’s theorems muddy the waters between where objective truth ends and invented math begins.

拉蒂凯宁提出了一个很好的观点，即哥德尔定理混淆了客观真理结束和发明数学开始之间的水域。

One historical way people have tried to overcome the limitations of Gödel’s theorems has been to propose additional axioms beyond the commonly accepted ones.

人们试图克服哥德尔定理局限性的一种历史方法是提出超出普遍接受的公理的额外公理。

Say you want to prove a statement with the traditional axioms, but you find that you can’t — that it is undecidable.

假设你想用传统公理证明一个命题，但你发现你不能——它是不可判定的。

If you add a new axiom to your starting set, you may then be able to prove the statement true.

如果您向起始集合添加一个新的公理，您就可以证明该陈述是正确的。

Adding a different axiom, however, and you may be able to prove it false.

然而，添加一个不同的公理，你也许能够证明它是错误的。

So whether it’s true or false depends on the choice you’ve made.

所以它是真是假取决于你所做的选择。

Suddenly, “truth” is more contingent on one’s preferences or assumptions.

突然之间，“真相”变得更加取决于一个人的偏好或假设。

REBECCA GOLDSTEIN , philosopher and author of Incompleteness: The Proof and Paradox of Kurt Gödel

丽贝卡·戈德斯坦 (REBECCA GOLDSTEIN)，哲学家和《不完备性：库尔特·哥德尔的证明与悖论》一书的作者

Intuitions have always played an important role in mathematics.

直觉在数学中一直发挥着重要作用。

After all, we can’t prove everything; we need to accept some truths (i.e., the axioms) without proof in order to get our proofs off the ground.

毕竟，我们无法证明一切；我们需要接受一些没有证据的真理（即公理），以便让我们的证明落地。

But we’ve learned over the centuries that sometimes intuitions prove unreliable — so unreliable as to generate actual paradoxes — meaning we’re driven to assert out-and-out contradictions.

但几个世纪以来我们已经了解到，有时直觉被证明是不可靠的——不可靠到产生实际的悖论——这意味着我们被迫断言彻头彻尾的矛盾。

In the early 20th century, Bertrand Russell and Alfred North Whitehead were working on The Principles of Mathematics , which attempted to reduce arithmetic to logic.

20世纪初，伯特兰·罗素和阿尔弗雷德·诺斯·怀特海正在研究《数学原理》，试图将算术简化为逻辑。

[The view that math is nothing but logic is known as “logicism.”] The work led Russell to the discovery of what came to be called Russell’s Paradox.

[数学只不过是逻辑的观点被称为“逻辑主义”。]这项工作导致罗素发现了后来被称为罗素悖论的东西。

It concerns the set of all sets that aren’t members of themselves.

它涉及不属于其自身成员的所有集合的集合。

The paradox reveals itself when you ask: Is this set a member of itself?

当你问：这个集合是它自己的成员时，这个悖论就会显现出来。

The contradiction: If it is, then it isn’t.

矛盾的是：如果是，那就不是。

And if it isn’t, then it is.

如果不是，那就是。

(Georg Cantor, considered the founder of set theory, had already realized the contradiction back in the 1890s.)

（被认为是集合论创始人的乔治·康托 (Georg Cantor) 早在 1890 年代就已经意识到了这一矛盾。）

The response of mathematicians — most forcefully David Hilbert , the leading mathematician of that time — was to rid mathematics of iffy intuitions by way of formally axiomatizing mathematics into a consistent and complete set of algorithmic, recursive rules, essentially reducing math to a mechanical game of symbol manipulation.

数学家——最有力的是当时的顶尖数学家戴维·希尔伯特——的反应是通过将数学形式上公理化为一套一致且完整的算法、递归规则，从本质上将数学简化为符号操作的机械游戏，从而消除数学中不确定的直觉。

This goal of formalization was christened the Hilbert Program.

这种形式化的目标被命名为希尔伯特计划。

What Gödel proved was that the Hilbert Program was unrealizable.

哥德尔证明的是希尔伯特纲领是无法实现的。

His first incompleteness theorem states that in every formal system of mathematics that is rich enough to express arithmetic, there will be propositions that are both true and unprovable.

他的第一个不完备性定理指出，在每一个足以表达算术的形式数学系统中，都会存在既真实又无法证明的命题。

So, although formal systems comprised of mechanical rules of symbol manipulation successfully eliminate all intuitions, they also fail to capture all that we know to be mathematically true — a knowledge enriched by intuitions concerning the infinite structures that we call numbers.

因此，尽管由符号操纵的机械规则组成的形式系统成功地消除了所有直觉，但它们也未能捕获我们所知道的所有数学上真实的知识——通过关于我们称为数字的无限结构的直觉丰富的知识。

It’s fascinating that our intuitions about numbers might go beyond what we can prove.

令人着迷的是，我们对数字的直觉可能超出我们所能证明的范围。

Personally, my intuition is silent on the mathematical statement that, in the years after Gödel’s proof, made incompleteness real.

就我个人而言，我的直觉对数学陈述保持沉默，在哥德尔证明之后的几年里，这个数学陈述使不完备性成为现实。

It is called the continuum hypothesis, and it asserts that the set of all real numbers (the continuum) is the second-smallest infinite set after the set of natural numbers (1, 2, 3 …).

它被称为连续统假设，它断言所有实数的集合（连续统）是继自然数集合（1,2,3…）之后的第二小的无限集合。

It was found to be undecidable using the standard axioms of mathematics.

人们发现使用数学标准公理它是不可判定的。

Extra axioms can be engineered to establish it as true or false, but logicians are divided on which way to go.

可以设计额外的公理来确定其真假，但逻辑学家对于该走哪条路存在分歧。

A physicist I spoke with warns that the undecidability of the continuum hypothesis has implications for his field: that physicists might need to avoid the continuum altogether.

与我交谈的一位物理学家警告说，连续统假设的不可判定性对其领域产生了影响：物理学家可能需要完全避免连续统。

CLAUS KIEFER , physicist at the University of Cologne, author of a 2024 paper on the relevance of Gödelian incompleteness for fundamental physics

CLAUS KIEFER，科隆大学物理学家，2024 年发表的一篇关于哥德尔不完备性与基础物理相关性论文的作者

Kurt Gödel’s proof has far-reaching and unexpected consequences for mathematics.

库尔特·哥德尔的证明对数学产生了深远且意想不到的影响。

Given that physical laws are formulated in mathematical language, is it relevant for physics, too?

鉴于物理定律是用数学语言表述的，它也与物理学相关吗？

I think yes.

我想是的。

Among the most important undecidable statements is the continuum hypothesis (CH), proved to be undecidable in the Gödelian sense by Paul Cohen in 1963.

最重要的不可判定陈述之一是连续统假设 (CH)，保罗·科恩 (Paul Cohen) 于 1963 年证明了它在哥德尔意义上是不可判定的。

The name “continuum” comes from the postulate to identify the points on a line with the real numbers.

“连续统”这个名字来自于用实数来识别直线上的点的假设。

But how many real numbers are there?

但有多少个实数呢？

There is an uncountable infinity of them, but can this uncountability be specified?

它们有不可数无穷，但是这个不可数可以指定吗？

The CH states that the real numbers form the next-smallest infinite set after the infinite set of the natural numbers, which are countable .

CH 指出，实数形成自然数无限集之后的下一个最小无限集，自然数是可数的。

Now consider that the known fundamental interactions in physics are defined on a space-time continuum.

现在考虑物理学中已知的基本相互作用是在时空连续体上定义的。

The uncountable number of points associated with this continuum is responsible for various problems in physics.

与这个连续体相关的无数点导致了物理学中的各种问题。

In Einstein’s theory of general relativity, for instance, our modern theory of gravity, it leads to singularities that prohibit the mathematical description of the universe’s origin and the interior of black holes.

例如，在爱因斯坦的广义相对论和我们现代的引力理论中，它导致了奇点，从而阻碍了对宇宙起源和黑洞内部的数学描述。

In the Standard Model of particle physics , described by a quantum field theory, direct calculations yield infinite results for energies and other physical quantities, which must be eliminated by a sophisticated and nonintuitive mathematical procedure.

在量子场论描述的粒子物理标准模型中，直接计算会产生能量和其他物理量的无限结果，必须通过复杂且非直观的数学程序来消除这些结果。

The situation becomes more severe in the push for a final unified theory of all interactions.

在推动所有相互作用的最终统一理论的过程中，情况变得更加严峻。

A unified theory should be characterized by a consistent and complete mathematical language.

统一的理论应该具有一致和完整的数学语言。

But if a unified theory were to describe space-time as a continuum, the CH may render the theory incomplete.

但如果一个统一理论将时空描述为一个连续体，那么 CH 可能会使该理论变得不完整。

Physicists have already shown that the CH leads to undecidable questions in quantum field theory, such as whether certain atomic systems have an “energy gap,” enabling them to settle into stable ground states.

物理学家已经证明，CH 会导致量子场论中无法判定的问题，例如某些原子系统是否具有“能隙”，使它们能够稳定在稳定的基态。

This undecidability stems from the fact that the calculation assumes the atoms inhabit a space-time continuum.

这种不可判定性源于这样一个事实：计算假设原子居住在时空连续体中。

One may argue that a more fundamental theory (with more complete axioms) could decide the question, but the final theory should not have undecidable statements.

有人可能会争辩说，更基本的理论（具有更完整的公理）可以决定问题，但最终的理论不应该有不可判定的陈述。

So it should not involve a continuum.

所以它不应该涉及连续体。

In my opinion, this situation of undecidability can only be avoided if the structure of space and time is discrete — that is, characterized by a countable infinity of points only.

在我看来，只有当空间和时间的结构是离散的（即仅以可数无穷个点为特征）时，才能避免这种不可判定性的情况。

There are hints for a discreteness in some approaches to quantum gravity, for example string theory or loop quantum gravity, but the situation is far from clear.

一些量子引力方法（例如弦理论或圈量子引力）中存在离散性的迹象，但情况还远未明朗。

It’s worth noting that on top of these troubles with the continuum hypothesis, high-energy physicists have many other reasons to think a continuous space-time is not fundamental to reality, but rather only a long-distance illusion that emerges from other parts.

值得注意的是，除了连续统假说的这些问题之外，高能物理学家还有许多其他理由认为连续时空并不是现实的基础，而只是从其他部分出现的长距离幻觉。

JOUKO VÄÄNÄNEN , mathematician and logician at the universities of Helsinki and Amsterdam

JOUKO VääNäNEN ，赫尔辛基大学和阿姆斯特丹大学数学家和逻辑学家

Incompleteness is an unwelcome but unavoidable fact of life in mathematics, like irrational and transcendental numbers in number theory, or Heisenberg’s uncertainty principle in physics.

不完备性是数学中一个不受欢迎但不可避免的事实，就像数论中的无理数和超越数，或者物理学中的海森堡测不准原理一样。

There is a kind of “Gödel barrier” that formal language cannot circumvent: The stronger the expressive power of a logic (meaning the more things you can say in the logic), the weaker is its effectiveness (meaning our ability to prove statements true or false in the logic), and the stronger the effectiveness, the weaker is the expressive power.

有一种形式语言无法规避的“哥德尔障碍”：逻辑的表达力越强（即在逻辑中你能说的东西越多），其有效性就越弱（即我们在逻辑中证明陈述真假的能力），而有效性越强，表达力越弱。

For example, one of the simplest logical systems is propositional logic, which lets you combine statements with operations such as “and,” “or,” and “not.” It is very effective, but its expressive power is weak.

例如，最简单的逻辑系统之一是命题逻辑，它允许您将语句与“与”、“或”和“非”等运算结合起来。它很有效，但表现力较弱。

On the other end of the spectrum, there’s second-order logic, which lets you make statements about objects, properties, sets, and relationships.

另一方面，还有二阶逻辑，它可以让你对对象、属性、集合和关系做出陈述。

It has tremendous expressive power and very weak effectiveness.

它具有巨大的表现力，但有效性却很弱。

It is as if the “product” of effectiveness and expressive power were constant, just as in Heisenberg’s uncertainty principle, which says that there is a limit to the precision with which certain “complementary” pairs of physical properties, such as position and momentum, can be simultaneously known; in other words, the more accurately one property is measured, the less accurately the other property can be known.

就好像有效性和表达力的“乘积”是恒定的，就像海森堡的不确定性原理一样，该原理说，可以同时知道某些“互补”物理属性对（例如位置和动量）的精确度是有限的；换句话说，对一种属性的测量越准确，对另一种属性的了解就越不准确。

In logic, in a remarkable analogy, effectiveness and expressiveness are such “complementary” properties.

在逻辑中，有一个显着的类比，有效性和表现力就是这样的“互补”属性。

This is the real content of Gödel’s incompleteness theorems.

这就是哥德尔不完备定理的真正内容。

We stumble forward in mathematics without any certainty of consistency or completeness.

我们在数学上跌跌撞撞地前进，没有任何一致性或完整性的确定性。

This is just how things are.

事情就是这样。

It is shocking that mathematics, which is the basis of exact sciences, lacks a foundation that can be proved to be consistent and complete.

令人震惊的是，作为精确科学基础的数学却缺乏一个可以被证明是一致和完整的基础。

Hilbert can be forgiven for thinking that this cannot be the case.

希尔伯特认为情况并非如此，这是情有可原的。

However, it is the case, as certainly as the square root of two is irrational.

然而，情况确实如此，就像二的平方根是无理数一样。

Mathematics has a puzzling lump of incompleteness which can be pushed from place to place but it will never disappear.

数学有一个令人费解的不完整性，可以从一个地方推到另一个地方，但它永远不会消失。

Surprisingly, Gödel himself was a little more optimistic.

令人惊讶的是，哥德尔本人更为乐观一些。

Here, Rachael Alvir explains that Gödel maintained the dream of a formal logical system that could settle the continuum hypothesis and all other questions about sets, the building blocks of modern mathematics.

在此，雷切尔·阿尔维尔 (Rachael Alvir) 解释说，哥德尔一直梦想着建立一个形式逻辑系统，该系统可以解决连续统假设以及有关集合（现代数学的基石）的所有其他问题。

His incompleteness theorems tell us that any such system, so long as it consists of a finite list of axioms, will give rise to new statements that are undecidable within that system.

他的不完备性定理告诉我们，任何这样的系统，只要它由有限的公理列表组成，就会产生在该系统内不可判定的新陈述。

But he wondered about the possibility of an infinite succession of ever-larger axiomatic systems that could settle every question.

但他想知道是否有可能出现无限连续的更大的公理系统来解决所有问题。

RACHAEL ALVIR , logician and lecturer at the University of Waterloo

RACHAEL ALVIR ，滑铁卢大学逻辑学家兼讲师

We have all been exposed to the general idea that Gödel killed Hilbert’s Program for thorough formalization of math.

我们都知道哥德尔为了数学的彻底形式化而扼杀了希尔伯特纲领。

This is a common interpretation, so I was shocked when I first read Gödel’s original works.

这是一种常见的解释，所以当我第一次读到哥德尔的原作时，我感到很震惊。

In his 1931 paper , in which the incompleteness theorems are first proven, Gödel explicitly states the opposite: “It must be expressly noted that Proposition XI (and the corresponding results for M and A) represent no contradiction of the formalistic standpoint of Hilbert.” In a footnote, he reiterates that the undecidable theorems of the 1931 paper are only undecidable relative to one system.

哥德尔在 1931 年首​​次证明不完备性定理的论文中明确提出了相反的观点：“必须明确指出，命题 XI（以及 M 和 A 的相应结果）与希尔伯特的形式主义立场并不矛盾。”在脚注中，他重申 1931 年论文中的不可判定定理仅相对于一个系统是不可判定的。

The undecidable statements of any given logical framework can be mathematically proven to be true or false in a larger logical framework.

任何给定逻辑框架的不可判定的陈述都可以在更大的逻辑框架中用数学方法证明是对还是错。

Gödel had no qualm with the claim that mathematics could prove or disprove every well-posed statement.

哥德尔毫不犹豫地宣称数学可以证明或反驳每一个适定命题。

Rather, Gödel took issue with Hilbert’s restrictive methods.

相反，哥德尔对希尔伯特的限制性方法提出了质疑。

Why should we believe there is a single, finite set of axioms, from which every truth will follow in a finite number of logical steps?

为什么我们应该相信存在一个单一的、有限的公理集，每个真理都可以通过有限数量的逻辑步骤得出？

Gödel believed that it was possible to redefine what we mean by a formal mathematical framework, or allow for alternative frameworks.

哥德尔认为，可以重新定义正式数学框架的含义，或者允许替代框架。

He often discussed an infinite sequence of acceptable logical systems, each more powerful than the last.

他经常讨论可接受的逻辑系统的无限序列，每个系统都比上一个更强大。

Every well-formulated mathematical question might be answerable within one of them.

每一个精心设计的数学问题都可以在其中一个问题中找到答案。

Often people will speak as if the CH is the smoking gun that shows sometimes mathematical questions have no answer.

通常人们会说 CH 是确凿的证据，表明有时数学问题没有答案。

But in my opinion, this situation provides very little evidence that there are “absolutely undecidable” mathematical problems, relative to any given permissible framework.

但在我看来，这种情况几乎没有证据表明相对于任何给定的允许框架存在“绝对不可判定”的数学问题。

It is simply one example of a statement which has not currently been decided, and on its own provides no reason to suspect it could not be decided in the future using new techniques.

这只是目前尚未决定的声明的一个示例，并且其本身没有理由怀疑未来无法使用新技术来决定。

There are extensive, ongoing debates about this deep in the trenches of mathematics and philosophy.

在数学和哲学的深处，关于这个问题存在着广泛的、持续的争论。

The strongest point I wish to make is that the mathematical results, on their own, cannot settle the question.

我想说的最强烈的一点是，数学结果本身并不能解决问题。

It is far from obvious that there are mathematical questions with no solution.

存在无解的数学问题远非显而易见。

For me, Gödel’s theorems do not show that mathematics is limited, but rather that mathematics is much wider and more powerful than Hilbert’s finitistic view.

对我来说，哥德尔定理并不表明数学是有限的，而是表明数学比希尔伯特的有限论观点更广泛、更强大。

Alvir further clarified that there are different ways the old dream of mathematical truth might be realized.

阿尔维尔进一步澄清，可以通过多种方式实现数学真理的古老梦想。

One approach could be to tack on to the commonly accepted axioms a new one that settles the CH and doesn’t otherwise lead to any contradictions.

一种方法可能是在普遍接受的公理上添加一个新的公理，以解决 CH 并且不会导致任何矛盾。

Another approach is to discover a scheme for an infinitude of axioms that settles the CH and other questions.

另一种方法是发现解决 CH 和其他问题的无限公理的方案。

Or we could switch to a different logical system than the standard one, and in that alt-logic, settle the CH.

或者我们可以切换到与标准系统不同的逻辑系统，并在该替代逻辑中解决 CH。

(“My personal favorite [logical system] is called L-omega-1-omega,” Alvir told me, for anyone who wants to explore that further.) Or maybe the answer is “something totally new,” she said — “a truly novel stroke of creative genius.

（“我个人最喜欢的[逻辑系统]被称为L-omega-1-omega，”阿尔维尔告诉我，对于任何想要进一步探索它的人来说。）或者答案也许是“全新的东西，”她说——“真正新颖的创意天才之笔。

…

……

We come up with radically new mathematical techniques to solve problems all the time.

我们一直在想出全新的数学技术来解决问题。

Why expect we won’t do the same for the CH?”

为什么我们不会为 CH 做同样的事情呢？”

Of course, proving the CH true or false wouldn’t vanquish all undecidability.

当然，证明 CH 的真假并不能消除所有的不可判定性。

I’m going to let Väänänen’s colleague (and wife) have the last word.

我会让 Väänänen 的同事（和妻子）做出最后决定。

JULIETTE KENNEDY , philosopher of mathematics and mathematical logician at the University of Helsinki, editor of Interpreting Gödel: Critical Essays

朱丽叶·肯尼迪（JULIETTE KENNEDY），赫尔辛基大学数学哲学家、数理逻辑学家，《诠释哥德尔：批判论文》编辑

It is easy to lose one’s sense of wonder at the fact that such a blindingly obvious set of axioms — the Peano axioms for arithmetic (the set of rules about the natural numbers 0, 1, 2, 3 …

人们很容易对这样一组显而易见的公理失去好奇心——算术皮亚诺公理（关于自然数 0、1、2、3 ……的一组规则）。

closely related to the system that Gödel used in his proof, such as the rule, “Every number has a successor”) — is essentially incomplete and undecidable, meaning that all axiomatizable consistent extensions are incomplete and undecidable.

与哥德尔在证明中使用的系统密切相关，例如“每个数都有后继”规则）——本质上是不完整且不可判定的，这意味着所有可公理化的一致扩展都是不完整且不可判定的。

Hold on to that wonder!

坚持住这个奇迹！

The incompleteness theorems teach us that when it comes to our attempt to master the conceptual order, whether it be in mathematics or, for that matter, in any other domain, we will always fail — and indeed, in this case more than any other, we should be glad to have failed, for failure was clearly the more interesting, the more profound, outcome.

不完备性定理告诉我们，当我们试图掌握概念秩序时，无论是在数学领域，还是在任何其他领域，我们总是会失败——事实上，在这种情况下，我们比任何其他领域都更应该为失败而感到高兴，因为失败显然是更有趣、更深刻的结果。

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《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。

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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。