内容来源：https://www.quantamagazine.org/what-do-godels-incompleteness-theorems-truly-mean-20260518/

内容总结：

哥德尔不完备定理的真正含义：数学真理的边界与未解之谜

1931年，年仅25岁的奥地利逻辑学家库尔特·哥德尔通过将逻辑反躬自省的方式，提出了两项彻底改变知识与真理图景的定理。这些“不完备定理”表明：任何足以表达算术的形式化数学体系——即任何一套有限的公理或规则——都无法做到完备。总有一些真实的数学陈述无法从这些公理中逻辑推导出来。

新冠疫情期间，笔者花了许多周时间研习这位年轻逻辑学家的证明，并尝试用不到2000字的篇幅作出解释。但即便理解了证明步骤，哥德尔定理究竟意味着什么，仍然令人困惑。60多年过去，哲学家、数学家乃至物理学家对此仍有诸多争论。

不完备定理的核心意义

芬兰坦佩雷大学哲学家帕努·拉蒂凯宁指出，自古希腊以来，公理法一直被视作组织科学知识的理想方式，即用少量“自明”的基本命题推导出所有真理。哥德尔的不完备定理用数学的精确性证明，这一理想在数学的大部分领域必然失效。即便是针对正整数的全部数学真理，其复杂程度也远超任何有限公理体系的覆盖范围。这意味着，有些数学问题甚至在原则上都无法用现有方法解决，进展可能依赖于创造性的概念革新。

直觉与形式化的博弈

哲学家兼作家丽贝卡·戈德斯坦回顾了20世纪初罗素与怀特海的逻辑主义尝试。罗素悖论的发现揭示了直觉的不可靠性，促使数学界领军人物希尔伯特提出“希尔伯特计划”，试图通过形式化将数学变成一套机械化的符号游戏。哥德尔证明这一计划不可实现：任何足以表达算术的形式系统，都存在既真实又无法证明的命题。有趣的是，人类对数字的直觉反而可能超越形式证明的能力。

连续统假设与物理学困境

连续统假设——断言实数集是继自然数集之后第二小的无限集——被证明在标准公理下无法判定。科隆大学物理学家克劳斯·基弗警告，这一未决问题对基础物理学有深远影响。当前物理定律建立在时空连续统之上，但连续统中不可数多的点导致了广义相对论中的奇点、量子场论中的无穷大等问题。基弗认为，一个完备的终极统一理论不应包含连续统，暗示时空结构本身可能是离散的。

“哥德尔屏障”与互补性

赫尔辛基大学数学家约科·韦内宁提出，逻辑存在类似海森堡测不准原理的“互补性”：表达能力越强的逻辑体系，其有效性越弱；有效性越强的体系，表达能力越弱。这种“哥德尔屏障”无法绕过。数学就像带着不完备的“疙瘩”前行，可以把它推到别处，却永远无法彻底消除。

哥德尔的乐观面：并非终结，而是更广阔的数学

滑铁卢大学逻辑学家雷切尔·阿尔维尔指出，哥德尔本人并未认为自己的定理宣判了数学的终结。恰恰相反，哥德尔明确表示，不完备定理与希尔伯特的立场并不矛盾。他认为，通过无限递升的公理体系序列，每一个表述良好的数学问题终将在某个体系中得到解答。连续统假设之所以尚未解决，只是因为我们还没有找到合适的新公理或新技术。“数学比希尔伯特的有限主义观点要广阔和强大得多，”阿尔维尔强调。

最后的思考

赫尔辛基大学数学哲学家朱丽叶特·肯尼迪总结道，看似再明白不过的皮亚诺算术公理体系本质上就是不完备且不可判定的——所有可公理化的相容扩展也是如此。她感叹：正是不完备定理教会我们，无论是数学还是其他任何领域，在试图掌握概念秩序的过程中，我们永远会遭遇失败。“我们应当庆幸这种失败，因为失败显然带来了更深刻、更有趣的结果。”

中文翻译：

哥德尔不完备定理的真正含义是什么？
1931年，库尔特·哥德尔通过将逻辑学应用于自身，证明了两条定理，彻底改变了知识与真理的格局。这两条“不完备定理”确立了：任何数学形式系统——即任何有限条规则或公理构成的、可推演出一切结论的体系——都不可能完备。总存在一些真实的数学陈述，无法从这些公理中逻辑推导出来。

新冠疫情期间，我花了几周时间，研究这位25岁的奥地利逻辑学家兼数学家是如何做到这一点的，随后用不到2000字概述了他的证明过程。（当我向妻子提起这段经历时，她说：“哦，就是那次你差点疯掉的时候？”有点夸张。）

然而，即便理解了哥德尔证明的步骤，我仍不确定该如何看待他的定理——人们通常认为，这些定理排除了建立数学“万有理论”的可能性。并非只有我如此困惑。在《哥德尔的证明》（一本1958年的经典著作，我的概述很大程度上依赖于此书）中，哲学家欧内斯特·内格尔和数学家詹姆斯·R·纽曼写道，哥德尔定理的含义“尚未被充分理解”。

或许如此，但自那时起已过去六十年。如今我们对这些思想的理解到了哪一步？最近，我请教了逻辑学家、数学家、哲学家，还有一位物理学家，请他们探讨不完备定理的含义。关于哥德尔这一奇特智力成就的影响，以及它如何改变了人类永不停歇的真理探索之路，他们有很多话要说。

帕努·拉蒂凯宁，坦佩雷大学哲学家，《斯坦福哲学百科全书》哥德尔不完备定理条目的作者
自古希腊以来，公理方法就被广泛视为组织科学知识的理想方式。其目标是建立少数几条“不言自明”的基本命题——公理、原则或定律——使得该领域的所有真理都能从它们中逻辑推导出来。
哥德尔的不完备定理以数学精度表明，对于数学的大部分内容，这一理想必然失败。即使仅针对正整数（1, 2, 3……）的数学真理，其复杂性也令人困惑，以至于无法从任何有限公理集中推导出来。
这意味着，有些数学问题，即便在原则上，也无法用我们当前的数学方法解决。进步可能需要创造性的概念创新。因此，数学真理并非由同等确凿无疑的真理构成的统一整体；相反，它们作为知识的地位，从无可置疑的事实逐渐过渡到越来越不确定的假设。

拉蒂凯宁提出了一个很好的观点：哥德尔的定理模糊了客观真理的终点与人为发明的数学的起点之间的界限。历史上，人们试图克服哥德尔定理局限性的一种方式，是在普遍接受的公理之外提出额外的公理。假设你想用传统公理证明某个陈述，但发现无法证明——即该陈述不可判定。如果你在起始公理集中添加一条新公理，或许就能证明该陈述为真。然而，如果添加一条不同的公理，你或许又能证明它为假。因此，它是真是假取决于你的选择。突然间，“真理”更多地取决于个人的偏好或假设。

丽贝卡·戈尔茨坦，哲学家，《不完备：库尔特·哥德尔的证明与悖论》的作者
直觉在数学中一直扮演着重要角色。毕竟，我们无法证明一切；为了让证明能够进行，我们需要不加证明地接受某些真理（即公理）。但几个世纪以来，我们认识到，有时直觉并不可靠——甚至不可靠到产生实际悖论的地步——这意味着我们被迫断言彻头彻尾的矛盾。
20世纪初，伯特兰·罗素和阿尔弗雷德·诺思·怀特海正在撰写《数学原理》，试图将算术还原为逻辑。[认为数学不过是逻辑的观点被称为“逻辑主义”。]这项工作使罗素发现了后来被称为“罗素悖论”的问题。它涉及所有不属于自身的集合的集合。当你问：这个集合是否属于它自身？悖论便显现出来：如果属于，则不属于；如果不属于，则属于。（被认为创立了集合论的格奥尔格·康托尔，早在19世纪90年代就已意识到这个矛盾。）
数学家们的回应——尤其是当时顶尖数学家大卫·希尔伯特——是通过将数学形式化为一套一致且完备的算法化、递归规则，从而消除数学中不可靠的直觉，本质上将数学简化为一种机械的符号操作游戏。这一形式化目标被称为“希尔伯特纲领”。
哥德尔证明的是，希尔伯特纲领无法实现。他的第一不完备定理指出，在每个足够丰富、能够表达算术的数学形式系统中，都存在既真实又不可证明的命题。因此，尽管由机械符号操作规则构成的形式系统成功消除了所有直觉，但它们也未能捕捉到我们所知的全部数学真理——而这一知识正是由关于我们称之为数的无限结构的直觉所丰富。

我们对数的直觉可能超越我们所能证明的范围，这一点令人着迷。
就我个人而言，我对那条在哥德尔证明之后使不完备性成为现实的数学陈述毫无直觉。它被称为连续统假设，断言所有实数构成的集合（连续统）是仅次于自然数集（1, 2, 3……）的第二小无限集。人们发现，使用数学的标准公理无法判定其真伪。可以设计额外的公理来确立其为真或为假，但逻辑学家们对于选择哪条路径存在分歧。
一位与我交谈的物理学家警告说，连续统假设的不可判定性对其领域有影响：物理学家可能需要完全避开连续统。

克劳斯·基弗，科隆大学物理学家，2024年发表了一篇关于哥德尔不完备性与基础物理学相关性的论文
库尔特·哥德尔的证明对数学具有深远且出乎意料的影响。鉴于物理定律是用数学语言表述的，它是否也与物理学相关？我认为是的。
最重要的不可判定陈述之一是连续统假设（CH），1963年保罗·科恩证明它在哥德尔意义上是不可判定的。“连续统”这一名称源于将直线上的点等同于实数的假设。但实数有多少个？它们有无穷多个不可数，但能否明确这种不可数性呢？CH指出，实数构成比可数自然数集大的下一个最小无限集。
现在考虑，物理学中已知的基本相互作用都是在时空连续统上定义的。与这个连续统相关的不可数个点导致了物理学中的各种问题。例如，在爱因斯坦的广义相对论（我们现代的引力理论）中，这导致了奇点，使得无法对宇宙起源和黑洞内部进行数学描述。在由量子场论描述的粒子物理标准模型中，直接计算会得出能量和其他物理量的无穷大结果，必须通过复杂且非直观的数学过程加以消除。
在寻求所有相互作用的最终统一理论时，情况变得更加严峻。一个统一的理论应该以一致且完备的数学语言为特征。但如果一个统一理论将时空描述为连续统，CH可能使该理论变得不完备。物理学家已经证明，CH会导致量子场论中出现不可判定的问题，例如某些原子系统是否具有“能隙”，使其能够进入稳定的基态。这种不可判定性源于计算假设原子存在于时空连续统中。有人可能会争辩说，一个更基础的理论（具有更完备的公理）可以判定这个问题，但最终理论不应包含不可判定的陈述。因此，它不应涉及连续统。
在我看来，只有当时空结构是离散的——即仅由可数无限个点构成——时，才能避免这种不可判定性。某些量子引力方法（如弦理论或圈量子引力）中暗示了离散性，但情况远未明朗。

值得注意的是，除了连续统假设带来的这些麻烦，高能物理学家还有许多其他理由认为，连续的时空并非现实的基础，而只是一种长距离的幻想，从其他部分涌现出来。

尤科·韦内宁，赫尔辛基大学和阿姆斯特丹大学的数学家兼逻辑学家
不完备性是数学中一个不受欢迎却不可避免的事实，就像数论中的无理数和超越数，或物理学中的海森堡测不准原理一样。
存在一种“哥德尔屏障”，形式语言无法绕过：逻辑的表达力越强（意味着你能在该逻辑中言说更多事物），其有效性就越弱（意味着我们证明陈述在逻辑中真伪的能力）；而有效性越强，表达力就越弱。
例如，最简单的逻辑系统之一是命题逻辑，它允许你使用“与”、“或”、“非”等运算组合陈述。它非常有效，但表达力很弱。在光谱的另一端，存在二阶逻辑，它允许你针对对象、性质、集合和关系做出陈述。它具有极强的表达力，但有效性很弱。仿佛有效性与表达力的“乘积”是常数，就像海森堡测不准原理一样，该原理指出，某些“互补”的物理性质对（如位置和动量）不可能同时被精确知晓；换句话说，一个性质测量得越准确，另一个性质能获知的准确度就越低。在逻辑学中，惊人地类似，有效性和表达力就是这样的“互补”性质。这就是哥德尔不完备定理的真正内涵。
我们在数学中蹒跚前行，对一致性或完备性毫无把握。现实就是如此。

令人震惊的是，作为精密科学基础的数学，其本身却缺乏一个能被证明为一致且完备的基础。希尔伯特认为情况不应如此，这是可以理解的。然而，事实确实如此，就像根号2是无理数一样确定无疑。数学中存在一块令人困惑的不完备之瘤，可以被从一个地方推到另一个地方，但它永远不会消失。

令人惊讶的是，哥德尔本人反倒更乐观一点。拉谢尔·阿尔维尔解释说，哥德尔始终怀有一个梦想：建立一个形式逻辑系统，能够判定连续统假设以及关于集合（现代数学的构建基石）的所有其他问题。他的不完备定理告诉我们，任何这样的系统，只要由有限条公理构成，就会产生在该系统内不可判定的新陈述。但他思考过是否存在一种可能性，即一个由越来越庞大的公理系统构成的无限序列，能够判定每一个问题。

拉谢尔·阿尔维尔，滑铁卢大学逻辑学家兼讲师
我们所有人都接触过这样一个普遍观点：哥德尔扼杀了希尔伯特彻底形式化数学的纲领。这是一种常见的解读，因此当我第一次阅读哥德尔的原著时，感到震惊。在1931年首次证明不完备定理的那篇论文中，哥德尔明确指出了相反的观点：“必须特别指出，命题XI（以及针对M和A的相应结果）并不代表与希尔伯特形式主义立场的矛盾。”在一个脚注中，他重申，1931年论文中的不可判定定理仅仅是相对于一个系统而言的不可判定。任何给定逻辑框架下的不可判定陈述，都可以在更大的逻辑框架中通过数学证明其为真或为假。
哥德尔并不反对数学能够证明或证伪每一个表述得当的陈述这一说法。相反，哥德尔质疑的是希尔伯特的限制性方法。为什么我们要相信存在一个单一的、有限的公理集，所有真理都能从中以有限数量的逻辑步骤推导出来？哥德尔相信，重新定义我们所谓的正式数学框架，或者允许替代框架，是可能的。他经常讨论一个可接受的逻辑系统的无限序列，每个系统都比前一个更强大。每一个表述得当的数学问题都可能在其中某个系统中得到解答。
人们常常会谈论连续统假设，仿佛它是确凿的证据，表明有时数学问题没有答案。但在我看来，这种情况提供的证据非常薄弱，不足以证明存在相对于任何给定允许框架“绝对不可判定”的数学问题。它仅仅是一个目前尚未被判定的陈述的例子，单凭它本身，没有理由怀疑未来无法使用新技术加以判定。在数学和哲学的深处，关于这一点存在着广泛且持续的辩论。
我最想强调的一点是，数学结果本身无法解决这个问题。是否存在没有解的数学问题，这一点远非显而易见。对我来说，哥德尔的定理并未表明数学是有限的，而是表明数学远比希尔伯特有限主义视角下的观点更为广阔和强大。

阿尔维尔进一步澄清，实现数学真理这一古老梦想有不同的方式。一种方法是在普遍接受的公理上附加一条能判定连续统假设且不会导致矛盾的新公理。另一种方法是发现一个包含无穷多公理的方案，以判定连续统假设及其他问题。或者，我们可以切换到一个不同于标准逻辑的逻辑系统，并在那种另类逻辑中判定连续统假设。（“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，供有兴趣深入了解的人参考。）又或者，答案可能是“某种全新的东西”，她说——“真正新颖的创造性天才之举……我们一直都在想出全新的数学技术来解决问题。为什么认为我们不会对连续统假设采取同样的做法呢？”

当然，证明连续统假设的真伪并不会消除所有不可判定性。
我打算让韦内宁的同事（也是妻子）来作总结。

朱丽叶·肯尼迪，赫尔辛基大学数学哲学学者兼数理逻辑学家，《解读哥德尔：批判性论文集》编辑
人们很容易对这样一个事实失去惊奇感：一套看似如此显而易见的公理——皮亚诺算术公理（关于自然数0, 1, 2, 3……的规则集，与哥德尔证明中使用的系统密切相关，例如“每个数都有一个后继”这条规则）——本质上是不完备且不可判定的，这意味着所有可公理化的、一致的扩展都是不完备且不可判定的。请保持那份惊奇！不完备定理告诉我们，当我们试图掌控概念秩序时，无论是在数学中，还是在任何其他领域，我们总会失败——而事实上，在这种情况下，比任何其他情况都更应该庆幸我们的失败，因为失败显然是更有趣、更深刻的结果。

英文来源：

What Do Gödel’s Incompleteness Theorems Truly Mean?
In 1931, by turning logic on itself, Kurt Gödel proved a pair of theorems that transformed the landscape of knowledge and truth. 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. There will always be true mathematical statements that don’t logically follow from those axioms.
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. (My wife, when I reminded her of this period: “Oh yeah, that time you almost went crazy?” A slight exaggeration.)
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. In Gödel’s Proof (a classic 1958 book that I heavily relied upon for my account), philosopher Ernest Nagel and mathematician James R. Newman wrote that the meaning of Gödel’s theorems “has not been fully fathomed.”
Maybe not, but six decades have passed since then. Where are we with these ideas today? Recently, I asked logicians, mathematicians, philosophers, and one physicist to discuss the meaning of incompleteness. 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.
PANU RAATIKAINEN, philosopher at Tampere University and author of the Stanford Encyclopedia of Philosophy entry on Gödel’s incompleteness theorems
Ever since the ancient Greeks, the axiomatic method has been widely taken as the ideal way of organizing scientific knowledge. 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.
Gödel’s incompleteness theorems show with mathematical precision that this ideal necessarily fails for large parts of mathematics. 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.
This means that some mathematical problems are not even in principle solvable by our current mathematical methods. Progress may require creative conceptual innovation. 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.
Raatikainen makes a good point that Gödel’s theorems muddy the waters between where objective truth ends and invented math begins. 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. Say you want to prove a statement with the traditional axioms, but you find that you can’t — that it is undecidable. If you add a new axiom to your starting set, you may then be able to prove the statement true. Adding a different axiom, however, and you may be able to prove it false. So whether it’s true or false depends on the choice you’ve made. Suddenly, “truth” is more contingent on one’s preferences or assumptions.
REBECCA GOLDSTEIN, philosopher and author of Incompleteness: The Proof and Paradox of Kurt Gödel
Intuitions have always played an important role in mathematics. 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. 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.
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. [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. It concerns the set of all sets that aren’t members of themselves. The paradox reveals itself when you ask: Is this set a member of itself? The contradiction: If it is, then it isn’t. And if it isn’t, then it is. (Georg Cantor, considered the founder of set theory, had already realized the contradiction back in the 1890s.)
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. This goal of formalization was christened the Hilbert Program.
What Gödel proved was that the Hilbert Program was unrealizable. 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. 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.
It’s fascinating that our intuitions about numbers might go beyond what we can prove.
Personally, my intuition is silent on the mathematical statement that, in the years after Gödel’s proof, made incompleteness real. 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 …). It was found to be undecidable using the standard axioms of mathematics. Extra axioms can be engineered to establish it as true or false, but logicians are divided on which way to go.
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.
CLAUS KEIFER, physicist at the University of Cologne, author of a 2024 paper on the relevance of Gödelian incompleteness for fundamental physics
Kurt Gödel’s proof has far-reaching and unexpected consequences for mathematics. Given that physical laws are formulated in mathematical language, is it relevant for physics, too? I think yes.
Among the most important undecidable statements is the continuum hypothesis (CH), proved to be undecidable in the Gödelian sense by Paul Cohen in 1963. The name “continuum” comes from the postulate to identify the points on a line with the real numbers. But how many real numbers are there? There is an uncountable infinity of them, but can this uncountability be specified? The CH states that the real numbers form the next-smallest infinite set after the infinite set of the natural numbers, which are countable.
Now consider that the known fundamental interactions in physics are defined on a space-time continuum. The uncountable number of points associated with this continuum is responsible for various problems in physics. 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. 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.
The situation becomes more severe in the push for a final unified theory of all interactions. A unified theory should be characterized by a consistent and complete mathematical language. But if a unified theory were to describe space-time as a continuum, the CH may render the theory incomplete. 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. This undecidability stems from the fact that the calculation assumes the atoms inhabit a space-time continuum. 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. So it should not involve a continuum.
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. 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.
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.
JOUKO VÄÄNÄNEN, mathematician and logician at the universities of Helsinki and Amsterdam
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.
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.
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. On the other end of the spectrum, there’s second-order logic, which lets you make statements about objects, properties, sets, and relationships. It has tremendous expressive power and very weak effectiveness. 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. In logic, in a remarkable analogy, effectiveness and expressiveness are such “complementary” properties. This is the real content of Gödel’s incompleteness theorems.
We stumble forward in mathematics without any certainty of consistency or completeness. This is just how things are.
It is shocking that mathematics, which is the basis of exact sciences, lacks a foundation that can be proved to be consistent and complete. Hilbert can be forgiven for thinking that this cannot be the case. However, it is the case, as certainly as the square root of two is irrational. Mathematics has a puzzling lump of incompleteness which can be pushed from place to place but it will never disappear.
Surprisingly, Gödel himself was a little more optimistic. 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. 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. But he wondered about the possibility of an infinite succession of ever-larger axiomatic systems that could settle every question.
RACHAEL ALVIR, logician and lecturer at the University of Waterloo
We have all been exposed to the general idea that Gödel killed Hilbert’s Program for thorough formalization of math. This is a common interpretation, so I was shocked when I first read Gödel’s original works. 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. The undecidable statements of any given logical framework can be mathematically proven to be true or false in a larger logical framework.
Gödel had no qualm with the claim that mathematics could prove or disprove every well-posed statement. Rather, Gödel took issue with Hilbert’s restrictive methods. 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? Gödel believed that it was possible to redefine what we mean by a formal mathematical framework, or allow for alternative frameworks. He often discussed an infinite sequence of acceptable logical systems, each more powerful than the last. Every well-formulated mathematical question might be answerable within one of them.
Often people will speak as if the CH is the smoking gun that shows sometimes mathematical questions have no answer. But in my opinion, this situation provides very little evidence that there are “absolutely undecidable” mathematical problems, relative to any given permissible framework. 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. There are extensive, ongoing debates about this deep in the trenches of mathematics and philosophy.
The strongest point I wish to make is that the mathematical results, on their own, cannot settle the question. It is far from obvious that there are mathematical questions with no solution. 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.
Alvir further clarified that there are different ways the old dream of mathematical truth might be realized. 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. Another approach is to discover a scheme for an infinitude of axioms that settles the CH and other questions. Or we could switch to a different logical system than the standard one, and in that alt-logic, settle the 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. … We come up with radically new mathematical techniques to solve problems all the time. Why expect we won’t do the same for the CH?”
Of course, proving the CH true or false wouldn’t vanquish all undecidability.
I’m going to let Väänänen’s colleague (and wife) have the last word.
JULIETTE KENNEDY, philosopher of mathematics and mathematical logician at the University of Helsinki, editor of Interpreting Gödel: Critical Essays
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 … 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. Hold on to that wonder! 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.