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哥德尔吉布斯篇：一些关于数学基础的基本定理及其意义（1951）

哲学园 · 公众号 · 哲学 · 2024-12-04 00:15

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一些关于数学基础的基本定理及其意义（1951）

哥德尔

在过去的几十年里，关于数学基础的研究已经产生了一些有趣的结果，这些结果不仅本身值得关注，而且对数学本质的传统哲学问题也有着深远的意义。我相信，这些结果已经被广泛知晓，但我仍然认为将它们概述一遍是有益的，尤其是因为得益于许多数学家的努力，这些结果比最初有了更为令人满意的形式。最大的改进源于对“有限程序”概念的精确定义，这在这些结果中起到了决定性的作用。有几种不同的方法可以达到这种定义，但它们都指向完全相同的概念。在我看来，最令人满意的方法是将有限程序的概念还原为一个具有有限部件的机器，这一点由英国数学家图灵所完成。关于这些结果的哲学意义，我认为它们从未被充分讨论过，或者仅仅被注意到。

我所考虑的元数学结果都围绕着一个基本事实，或者可以说是这个基本事实的不同方面，这个事实可以被称为数学的“不完备性”或“无穷性”。当公理化方法被应用时，这一事实在最简单的形式下得以显现，而不是应用于诸如几何学这样的假设演绎系统（在这些系统中，数学家只能断言定理的条件性真理），而是应用于真正的数学，即那些在绝对意义上成立、无需进一步假设的数学命题的体系。这种命题必须存在，否则连假设性的定理也无法存在。例如，某些形式的蕴含：

“如果假设某些公理，那么某个定理成立”，

这种命题必须在绝对意义上成立。同样地，有限主义数论中的任何定理，例如，无疑也属于这一类。当然，对真正数学进行公理化的任务与通常的公理化观念有所不同，因为这些公理并非任意，而必须是正确的数学命题，而且无需证明就显而易见。不可避免地需要假定某些公理或推理规则是无需证明而显而易见的，因为证明必须有一个起点。然而，对于我所定义的真正数学的扩展范围，各方的观点差异很大。例如，直觉主义者和有限主义者拒绝了一些公理和概念，而这些公理和概念是其他人承认的，比如排中律或集合的一般概念。

然而，无论采取何种立场，数学的无穷性现象总是以某种形式存在。因此，我不妨用最简单和最自然的立场来解释它，这种立场接受数学本身，而不对其进行任何批评。从这个立场看，所有数学都可以还原为抽象集合论。例如，投影几何的公理蕴涵某个定理，这意味着如果一个名为的元素集合（称为点）和一个名为的子集集合（称为直线）满足这些公理，那么这个定理对成立。再举一个例子，数论中的某个定理可以被解释为关于有限集合的断言。

由此，问题在于如何对集合论进行公理化。如果尝试解决这个问题，结果与预期会大不相同。与几何学不同，集合论不是以有限数量的公理结束，而是面对一个无限的公理序列，这些公理可以不断扩展，似乎没有尽头，而且显然不可能用一个有限的规则来生成所有这些公理。

这种情况的原因在于，如果希望避免集合论悖论而又不引入完全与实际数学程序无关的东西，就必须以逐步的方式对集合的概念进行公理化。例如，如果我们从整数（即一种特殊的有限集合）开始，首先有整数的集合及其公理（第一层次的公理），然后是整数集合的集合及其公理（第二层次的公理），依此类推，对“集合的集合”操作进行任何有限次数的迭代。接下来，我们有了所有这些有限次序集合的集合。但现在我们可以用处理整数集合的方式来处理这一集合，也就是说，考虑它的子集（即次序的集合）并为它们的存在形式化公理。

显然，这种程序可以超越进行迭代，实际上可以延伸到任意超限序数。因此，可能需要作为下一个公理规定，这种迭代对于任何序数（即属于某种良序集合的次序类型）都是可能的。但是我们现在到达尽头了吗？绝不是。我们现在有了一种新的形成集合的操作，即通过对某个初始集合和某个良序集合应用“集合的集合”操作，形成一个集合，然后将设为的某种良序，接着可以将这一新操作再次迭代到超限集合中。这将再次产生一个新的操作，我们可以用相同的方式处理，依此类推。因此，下一个步骤将要求任何生成集合的操作都可以迭代到任意序数（即某种良序集合的次序类型）。但是我们现在到达尽头了吗？不，因为我们还可以要求这种程序不仅适用于任何操作，而且还应当存在一个对其封闭的集合，即具有这样的性质：如果这一程序（包括任何运算）应用于该集合中的元素，那么结果仍然属于该集合。我认为你会意识到，我们还没有达到终点，也永远不会有一个终点，因为形成公理的这一过程本身就会在某个阶段导致下一个公理的产生。

的确，在当今的数学中，这一层次体系的较高层级几乎从未被使用。可以安全地说，99.9%的现代理论数学都包含在该层次体系的前三层中。 因此，出于所有实际目的，整个数学都可以归约为有限数量的公理。然而，这仅仅是一个历史偶然现象，对于原则性问题并没有什么重要意义。此外，这一现代理论数学的特性可能与它的另一个特性有关，即尽管经过多年的努力，它仍然无法证明某些基本定理，例如黎曼猜想。

事实上，可以证明高层次集合的公理在其相关性上并不仅限于这些集合，相反，甚至对0级的内容（即整数理论）也有影响。更确切地说，每一个这样的集合论公理都可以解决基于先前公理无法解决的某些丢番图问题。

这些丢番图问题是这样的类型：设为具有给定整数系数和个变量的多项式，，并将变量视为未知数，变量视为参数；问题是：对于参数的任何整数值，方程是否有整数解，或者是否存在参数的整数值使得该方程没有整数解？对于每个集合论公理，可以指定某个多项式，使得通过该公理，这一问题变得可判定。甚至可以总是保证的次数不高于4。

然而，当今的数学尚未学会使用集合论公理来解决数论问题，除了第一层次的公理。这些实际上在解析数论中被使用。但对于完全掌握数论，这显然是不够的。某种集合论数论（尚未被发现）无疑将能够达到更深远的领域。

到目前为止，我试图解释我所称为数学的不完备性，这一事实是关于数学基础的某种具体方法，即集合论公理化。然而，这一事实完全独立于所选的方法和立场，这一点可以从某些非常普遍的定理中得出。第一个定理简单地表明：无论选择何种定义明确的公理系统和推理规则，总会存在某些上述类型的丢番图问题，这些问题是这些公理和规则无法解决的，前提是不能从中推导出任何错误的命题。

在这里，如果我提到一个定义明确的公理和规则系统，这仅意味着必须能够用某种精确的形式写下这些公理，或者如果它们的数量是无限的，则必须给出一个有限的程序以依次写出它们。同样，推理规则也必须是这样的：给定任何前提，要么可以根据这些推理规则之一写下结论，要么可以确定在所考虑的推理规则下没有直接的结论。这一对规则和公理的要求等同于这样的要求：应当可以构建一个有限的机器（即严格意义上的“图灵机”），它可以一个接一个地写出公理的所有推论。因此，这一定理与这样一个事实是等价的：不存在一种有限的程序可以系统性地决定所有指定类型的丢番图问题。

第二个定理涉及无矛盾性的概念。对于定义明确的公理系统和规则，其一致性问题当然本身是一个定义明确的数学问题。此外，由于[任何]一种形式系统的符号和命题最多是可数的，因此一切都可以映射到整数上。可以论证并且事实上可以证明，一致性问题始终可以被转化为数论问题（更准确地说，是转化为上述类型的问题）。现在，这一定理表明：对于任何明确定义的公理系统和推理规则，尤其是陈述其一致性的命题（或更确切地说是等价的数论命题），如果这些公理和规则是一致的，并且足以推导出有限整数算术的某一部分，那么该命题是不可从这些公理和规则中证明的。正是这一定理使得数学的不完备性显得尤为明显。因为，它使得某人无法建立一个特定的、定义明确的公理和规则系统，并且关于该系统做出以下一致的断言：我以数学的确定性感知到这些公理和规则是正确的，而且我相信它们包含了所有的数学。如果有人作出这样的声明，他就自相矛盾。 因为如果他感知到所考虑的公理是正确的，他也以同样的确定性感知到它们是一致的。因此，他获得了一种不能从他的公理中推导出的数学洞见。然而，人们必须小心，以便清楚地理解这种状态的含义。这是否意味着没有任何定义明确的正确公理系统能够包含所有真正的数学内容？如果我们将数学内容理解为所有真正的数学命题的系统，那么答案是肯定的；但如果将其理解为所有可证明的数学命题的系统，那么答案是否定的。我将数学区分为客观意义和主观意义上的两种含义：

显然，任何定义明确的正确公理系统都无法包含所有客观数学，因为陈述该系统一致性的命题是真实的，但在该系统中不可证明。然而，关于主观数学，并不排除可能存在一种有限规则生成其所有明显的公理。然而，如果确实存在这样的规则，以我们人类的理解力，我们肯定永远无法确知它就是如此，也就是说，我们永远无法以数学的确定性知道它生成的所有命题都是正确的；换句话说，对于任何有限数量的命题，我们只能一个接一个地感知它们是真实的。然而，它们都是真实的这一断言，最多只能通过足够数量的实例或其他归纳推理以经验的确定性被知晓。如果是这样，这将意味着人类的心智（在纯数学领域内）等同于一台有限的机器，但这台机器却无法完全理解自己的运作方式。

这种人类[对自身的]理解能力的缺失可能会错误地被他视为其[心灵的]无限性或无穷性。但请注意，如果情况果真如此，这并不会削弱客观数学的不完备性。相反，它只会使其更为显著。因为，如果人类心智等同于一个有限机器，那么客观数学不仅在任何定义明确的公理系统中是不完备的，而且还会存在绝对无法解决的丢番图问题，如上所述，其中“绝对”的含义是指这些问题不仅在某些特定的公理系统内不可判定，而且超越了任何人类心智可构思的数学证明。因此，以下析取结论是不可避免的：

要么数学在这种意义上是不完备的，即其显然的公理永远无法包含在有限规则中，即便是在纯数学领域内，人类心智也远远超越了任何有限机器的能力；

要么存在上述类型的绝对不可解的丢番图问题（其中不排除析取命题的两部分同时为真，因此严格来说有三种可能）。

这一数学上确立的事实在我看来具有重大哲学意义。当然，在这个语境下，重要的是至少这一事实完全独立于数学基础所采取的特殊立场。

然而，这种独立性存在一个限制，即所采取的立场必须足够宽容，以承认关于所有整数的命题是有意义的。因为我们判断和对于任意两个整数与，，所依据的证据完全相同。此外，这种立场为了保持一致性，还必须排除那些涉及所有整数的概念，比如“+”（或涉及所有公式，例如“依据某些规则的正确证明”），并以仅适用于某些有限整数域（或公式域）的概念取而代之。然而需要注意的是，虽然这一析取定理的真实性独立于所采取的立场，但对于哪种选择成立的判断，则不必独立于这一立场。（见脚注[15]。）

然而，这种独立性存在一个限制，即所持立场必须足够宽容，能够承认关于所有整数的命题是有意义的。如果某人是如此严格的有限论者，以至于他坚持认为只有类似于的具体命题才属于真正的数学，那么不可完备性定理将不适用——至少，这个不可完备性定理不适用。但我认为，这种态度不可能一致地维持下去，因为我们判断和（对于任意两个整数）的依据完全是同一种证据。此外，为了一致，这种立场还必须排除涉及所有整数的概念，比如“+”（或者所有公式，比如“通过某种规则的正确证明”），并用仅适用于某些有限整数域（或公式域）的其他概念代替。然而需要注意的是，虽然析取定理的真理性与所持立场无关，但关于哪种选择成立的问题不一定独立于立场。（参见脚注[15]。）

我认为我已经充分解释了这种情形的数学方面，现在可以转向其哲学含义。当然，由于当今哲学的不发达状态，您不应该期望这些推论具有数学上的严谨性。

与数学不完备性定理的析取形式相对应，其哲学含义在初步看来也将是析取的；然而，无论选择哪种替代形式，它们都非常明确地反对唯物主义哲学。也就是说，如果第一种选择成立，这似乎意味着人类心智的运作不能还原为大脑的运作，而大脑在所有表象中是一种具有有限部分（即神经元及其连接）的有限机器。

因此，人们似乎不得不采取某种生命力论的观点。另一方面，第二种选择，即存在绝对不可判定的数学命题，似乎否定了数学仅是我们自己的创造物的观点；因为创造者必然知道其被造物的所有属性，因为它们除了创造者赋予的属性外不能拥有其他任何属性。

因此，这种观点似乎意味着数学对象和数学事实（或者至少它们的一些东西）以某种形式客观存在，并独立于我们的心智能力和决策。换句话说，这似乎暗示了某种形式的柏拉图主义或数学对象的“实在论”。

至于将数学解释为经验的观点，也就是说，认为数学事实是一种特殊的物理事实或心理事实的观点，过于荒谬，以至于无法被认真对待（见下文）。至于第一种选择是否成立，目前尚不确定，但无论如何，这一观点与一些脑神经生理学领域的权威人物的意见十分一致，这些人物明确否定了对心理和神经过程进行纯粹机械解释的可能性。

至于第二种选择，有人可能会反对说，构造者不一定必须知道其所构造物的每一个属性。例如，我们制造机器，但仍无法预测它们的每一个细节行为。然而，这种反对是站不住脚的。因为我们并不是凭空创造这些机器，而是用某种现成的材料构建它们。如果数学中的情况类似，那么我们构造的材料或基础将是某种客观的东西，即使数学的某些其他成分是我们自己的创造，这种材料也会迫使我们采取某种现实主义观点。如果我们在创造过程中使用了一种与自我不同的工具（例如“理性”，可以被解释为某种类似思维机器的东西），那么数学事实至少部分地会表现出这种工具的性质，而这种性质具有客观存在性。

第三种反对意见可能是，关于所有整数的命题的意义，由于无法逐一验证所有整数，只能在于一个一般性证明的存在。因此，对于一个关于所有整数的不可判定命题，其本身和其否定都不是真命题。因此，这两者都不表达整数的一个客观存在但未知的性质。

我目前无法讨论这一观点在认识论上是否一致。这确实表明，人必须先理解命题的意义，才能理解其证明，因此，“所有”的意义不能仅通过“证明”的意义来定义。但无论这一认识论探究如何，我想指出的是，人可以猜测某个全称命题的真实性（例如，我可以验证某个性质对于任何给定整数都成立），同时猜测对此事实不存在一般性证明。可以很容易想象出两种猜测都可能非常有依据的情况。其一，例如，当命题是关于某些数论函数的等式时，我们可以验证其在非常大的范围内的成立性。此外，正如在自然科学中，这种简单枚举归纳法远不是数学中可以想象的唯一归纳方法。我承认，每个数学家天生都不愿意赋予这种归纳论证更多的启发性意义。然而，我认为这源于一种偏见，即数学对象在某种程度上没有真实存在性。如果数学像物理学一样描述一个客观世界，就没有理由不在数学中应用与物理学相同的归纳方法。事实是，在数学中，我们今天仍然保持着过去对待所有科学的态度，即我们试图通过从定义（在本体论术语中，即事物的本质）中推导出有力的证明来推导出一切。如果这种方法声称垄断地位，那么它在数学中的错误性就如同它在物理学中的错误性一样。

这一整段论述表明，所阐述的数学事实的哲学意义并不完全倾向于理性主义或唯心主义哲学，而是在某种程度上更支持经验主义的观点。确实，只有第二种选择指向了这个方向。然而，这正是我现在想讨论的内容，在我看来，根据第二种选择得出的哲学结论，尤其是概念实在论（柏拉图主义），同样得到了现代数学基础发展所支持，而与选择哪种替代无关。

支持这一方向的主要论据似乎是以下几点。

首先，如果数学是我们自由创造的产物，那么对我们创造的对象的无知可能仍然会发生，但这仅仅是由于缺乏对我们实际上创造了什么的清晰认识（或者可能是由于过于复杂的计算导致的实际困难）。因此，这种无知必须（至少在原则上，尽管可能在实践中不一定）在我们达到完全清晰的时候消失。然而，现代数学基础发展的成就已经达到了难以超越的精确性，但这对于解决数学问题几乎没有实际帮助。

其次，数学家的活动显示出他享受的创造自由非常有限。即使例如关于整数的公理是自由发明的，但必须承认的是，数学家在想象了他的对象的最初几个性质之后，他的创造能力就已用尽，并且他也无法任意创造定理的有效性。如果数学中确实存在某种创造行为，那么任何定理的作用正是限制创造的自由。然而，这种限制显然必须独立于创造而存在。

第三，如果数学对象是我们的创造，那么显然整数和整数集必须是两种不同的创造，其中第一个并不必然需要第二个。然而，为了证明某些关于整数的命题，整数集的概念是必要的。因此，为了弄清楚我们赋予想象对象的性质，[我们]必须先创造其他某些对象——这确实是一个非常奇怪的情况！

到目前为止，我所说的内容是以“自由创造”或“自由发明”这个相当模糊的概念为基础进行表述的。有人尝试赋予这个术语更精确的含义。然而，这样做的唯一结果是，使得对相关立场的反驳也变得更加精确和有说服力。我想以最精确，同时也是目前为止最激进的表述为例，详细说明这一点。这种表述是将数学命题解释为仅仅表达语法（或语言）约定的某些方面，换句话说，数学命题仅仅是这些约定的部分内容的重复。根据这一观点，经过适当分析，数学命题的内容必然与“所有种马都是马”这样的陈述一样空洞。所有人都会同意，这一命题并不表达任何动物学或其他客观事实，而其真理性仅仅源于我们选择将“种马”作为“雄马”的简称这一约定。

目前最常见的符号约定类型是定义（可以是显式定义或情境定义，后者必须能够在任何出现其术语的上下文中消除被定义的术语）。因此，这种观点的最简单版本可以概括为这样一种主张：数学命题之所以为真，仅仅是因为它们中出现的术语的定义，即通过连续地用定义替换所有术语，任何定理都可以简化为明确的同义反复，。 (注意，如果定义被接受，就必须被承认为真，因为我们可以通过来定义，然后根据此定义，在该等式中用替换。）然而，根据前述定理，这种明确化为同义反复的简化是不可能的。因为它会立即产生一种机械程序，用于决定每个数学命题的真伪。然而，这种程序并不存在，即使是对于数论来说也不存在。

这一反驳确实只针对这种（唯名主义）观点的最简单版本。然而，更精细的版本也好不到哪去。为了使有关数学具有同义反复性质的观点具有可接受性，至少必须能证明以下最弱的陈述：每一个可证明的数学命题都可以仅根据关于句子真伪的规则推导出来（即，不需要使用或知道除了这些规则以外的任何其他东西），而且可证明数学命题的否定也不能由这些规则导出。

在精确表述的语言中，这些规则（即，规定在何种条件下一个给定的句子为真的规则）是用于确定句子意义的手段。此外，在所有已知的语言中，都有一些命题似乎仅凭这些规则为真。例如，如果用这些规则引入析取和否定：

当其至少有一个项为真时，为真；
如果不为真，则为真。

因此，可以从这些规则清楚地推导出，无论是什么，总是真命题。（通过这种方式推导出的命题被称为同义反复。）现在，实际上对于数学逻辑的符号系统，通过适当选择的语义规则，数学公理的真实性确实可以从这些规则中推导出来；然而（这是一个巨大的绊脚石），在这种推导中，数学和逻辑概念以及公理本身必须在一个特殊的应用中被使用，即，作为指代符号、符号的组合、此类组合的集合等。因此，这一理论如果想要证明数学公理的同义反复特性，必须首先假设这些公理是真实的。因此，尽管这种观点的最初目的是通过证明数学公理是同义反复来使其真实性变得可以理解，但最终结果却恰恰相反，即，必须首先假设公理为真，然后才能证明，在一个适当选择的语言中，它们是同义反复。

此外，类似的说法同样适用于数学概念，即，数学概念的意义不能仅通过符号约定来定义，人们必须首先了解它们的意义，才能理解相关的句法约定或证明它们暗示数学公理但不暗示其否定。现在，当然可以清楚地看到，这种对唯名主义观点的阐释并没有满足第[25?]页中提出的要求，因为不仅仅是句法规则，而是整个数学都被用在了推导中。但更进一步，这种唯名主义的阐释将彻底反驳这一观点（我必须承认，我无法想象比这一证明更好的对这一观点的反驳），前提是可以补充一个条件，即描述的结果是不可避免的（即，独立于所选择的具体符号语言和数学解释）。现在，这一点不能完全被证明，但有一些与之极为接近的内容，它同样足以反驳这一观点。即，根据前面提到的元定理，一个数学公理的同义反复性质的证明（在一种适当的语言中）同时也是它们一致性的证明，并且这种证明无法用比这些公理本身更弱的证明方法来实现。这并不意味着一个给定系统的所有公理都必须用于其一致性证明。相反，通常情况下，系统之外的必要公理可以使得某些系统内的公理变得不再需要（尽管这些外部公理并不蕴含系统内的公理）。然而，实际上可以肯定的是：为了证明经典数论（以及更强系统）的一致性，必须使用某些抽象概念（以及直接与它们相关的明显公理），其中“抽象”指的是不涉及感官对象的概念，符号则是其中一种特殊形式。然而，这些抽象概念显然不是句法化的[而是那些通过句法考虑来证明其合理性的概念，应成为唯名论的主要任务]。因此，可以得出结论：在句法解释的基础上，无法理性地证明我们关于经典数学（甚至它的最低层次，数论）适用性和一致性的前批判信念。 当然，这一说法并不适用于经典数学的某些子系统，这些子系统可能甚至包含所提到的抽象概念理论的一部分。从这个意义上说，唯名论可以指出一些部分成功的例子。例如，实际上可以基于纯句法的考虑来建立这些系统的公理。例如，通过句法考虑可以证明与整数相关的“所有”和“存在”概念的使用是合理的（即，被证明是一致的）。

然而，对于最基本的数论公理——完全归纳法，即使在句法基础可以适用的范围内，这种句法基础也无法为我们对它的前批判信念提供合理的证明，因为在句法性考量中，这一公理本身就必须被使用。

事实上，越是对公理持谨慎态度，即希望在设定一个逻辑冗余（如同直言命题）的解释时所使用的公理越少，则所需的数学量就越少。这导致的结果是，如果最终将自己限制在某个有限领域内，例如整数范围仅限于1000，那么在该领域内有效的数学命题可以被解释为逻辑冗余命题，甚至在最严格意义上也是如此，即通过术语的明确定义来还原为显式逻辑冗余。这并不令人惊讶，因为证明这种有限数学一致性所需的数学部分已经包含在有限组合过程的理论中，而该理论正是通过替换将公式还原为显式逻辑冗余所必需的。这解释了一个众所周知但却令人误解的事实：像这样的公式可以通过某些定义简化为显式逻辑冗余。这个事实偶然令人产生误解，因为如果这些还原被解释为在明确定义的基础上对定义项进行的简单替换，的含义并不等同于普通的加法，因为它仅适用于有限数量的参数（通过列举这些有限的情况）。另一方面，如果加法是在上下文中定义的，那么在证明时就必须已经使用了有限流形的概念。类似的循环性也出现在是逻辑冗余的证明中，因为其直观意义上的析取和否定显然出现在其中。

这一观点的实质是，并不存在所谓的数学事实，我们相信能表达数学事实的命题的真实性仅仅意味着（由于定义命题含义的相当复杂的规则，即决定给定命题在何种情况下为真的规则），这些命题中发生了一种语言的“空转”，也就是说，这些规则使它们无论事实如何都是真的。这样的命题可以被恰当地称为空洞的内容。现在实际上可以建立一种语言，其中数学命题在这种意义上是无内容的。唯一的问题是：

为了证明某些数学事实不存在，必须使用完全相同的数学事实（或同样复杂的其他数学事实）；
通过这种方法，如果将经验事实分为两部分，和，并且在中没有任何意义，可以构建一种语言，使表达的命题变得没有内容。而如果你的对手说：“你正在任意忽略某些可观察的事实”，可以回答：“你也在做同样的事情，例如对完全归纳法定律的处理，我之所以认为它是真实的，是基于我对整数概念的理解（即感知）。”

然而，在这种错误的数学真理理论中，有一个成分是完全正确的，实际上揭示了数学的真实本质。即，一个数学命题确实没有说明任何关于存在于时空中的物理或心理现实的内容，因为它的真实性已经完全取决于命题中术语的意义，而与实际世界中的事物无关。然而，错误在于，这些术语的意义（即它们所表示的概念）被断言为人造的，仅仅由语义惯例构成。我认为事实是，这些概念形成了一个独立的客观现实，我们无法创造或改变它们，只能感知和描述它们。

因此，数学命题虽然并未说明任何关于时空现实的内容，但它仍然可能具有非常稳固的客观内容，只要它涉及概念之间的关系即可。数学中非“重言式”关系的存在，尤其体现在以下事实中：对于数学的基本术语，必须假设一些公理，而这些公理绝不是重言式（即在任何意义上可以归约为），但它们仍然能够从所考虑的基本术语的含义中推导出来。

例如，关于整数集合概念的基本公理，或者更确切地说是公理模式，表明给定整数的一个明确性质（即带有整数变量的命题表达式），存在一个整数集合，其成员具有性质。然而，考虑到可能自身包含“整数集合”这一术语，我们实际上得到了一系列关于集合概念的相当复杂的公理。然而，正如上述结果所示，这些公理不能被简化为任何实质上更简单的形式，更不用说显式重言式了。这些公理确实由于“集合”这一术语的意义而有效——甚至可以说，它们表达了“集合”这一术语的核心意义——因此它们可以恰当地被称为“分析性的”。然而，用“重言式”来形容它们，即认为它们没有内容，却是完全不恰当的，因为即使是断言满足这些公理的集合概念（或者这些公理的一致性）的存在，也远非空洞，以至于在没有再次使用集合概念或其他类似的抽象概念的情况下无法证明这一点。

当然，这一特定论点仅针对那些在数学中承认集合一般概念的数学家。然而，对于有限主义者，完全相同的论点也可以用于整数概念和完全归纳公理。因为如果集合的一般概念在数学中不被接受，那么完全归纳必须被假设为一个公理。

我想再次强调，这里所说的“分析性”并不是指“由于我们的定义而为真”，而是指“由于所涉及概念的本质而为真”，与“由于事物的性质和行为而为真”形成对比。这种分析性概念绝不意味着“内容空洞”，因为完全可能存在某个分析命题是不可判定的（或仅能以[某种]概率判定）。因为我们对概念世界的认识可能和对事物世界的认识一样有限和不完整。这种认识在某些情况下显然是不完全的，甚至是不清晰的。这种情况出现在集合论的悖论中，这些悖论常常被用来作为反对柏拉图主义的证据，但我认为，这完全不公平。例如，当一根浸没在水中的杆视觉上与触觉感知相矛盾时，没有任何理性的人会因此得出外部世界不存在的结论。

我特意提到了两个独立的世界（事物的世界和概念的世界），因为我认为亚里士多德现实主义（根据其观点，概念是事物的部分或方面）不可取。

当然，我并不声称上述论述构成了关于数学本质这一观点的真实证明。我最多能够主张的是，已经驳斥了名义主义的观点，即认为数学仅仅由句法约定及其后果组成。此外，我还提出了一些有力的论据，反对数学是我们自身创造的这一更普遍的观点。然而，除了柏拉图主义之外，还存在其他替代理论，尤其是心理主义和亚里士多德现实主义。为了确立柏拉图式实在论，必须一个接一个地驳斥这些理论，并证明它们已穷尽了所有可能性。目前我还无法做到这一点；不过，我希望沿着这些方向给出一些提示。

心理主义的一种可能形式承认，数学研究的是概念之间的关系，而这些概念并非由我们的意志创造，而是作为一种我们无法改变的现实给予我们的；然而，它主张这些概念仅仅是心理倾向，也就是说，它们不过是我们思维机器的“齿轮”。更确切地说，一个概念可以被定义为以下倾向：

当我们想到它时会产生某种心理体验；

并且

对其与其他概念和经验对象的关系做出某种判断（或具有某种直接知识的经验）。

这种心理主义观点的本质是，数学的对象只是心理规律，即思想、信念等在我们身上发生的规律，就像心理学另一部分的对象是情绪在我们身上发生的规律一样。我目前看到对这一观点的主要反对意见是，如果它是正确的，那么我们将完全没有数学知识。例如，我们不会知道 2 + 2 = 4，只会知道我们的心智被构造为认为这是真的，并且不会有任何理由说明，为什么通过其他思路，我们不可能以同样程度的确定性得出相反的结论。因此，任何假设存在某种领域——无论多么小——的数学命题，其真理是我们已知的，都不能接受这一观点。*

我认为，在对所讨论的概念进行充分澄清之后，将有可能以数学的严格性进行这些讨论，并且结果将是（在某些几乎不可否认的假设下[特别是假设确实存在某种数学知识]），柏拉图主义的观点是唯一站得住脚的。我指的是这样一种观点：数学描述了一种非感官的现实，这种现实独立于人类心智的行为和倾向而存在，并且仅通过人类心智感知，而这种感知可能非常不完全。这种观点在数学家中相当不受欢迎；然而，确实有一些伟大的数学家支持这种观点。例如，埃尔米特曾写下以下句子：

"Il existe, si je ne me trompe, tout un monde qui est l’ensemble des vérités mathématiques, dans lequel nous n’avons accès que par l’intelligence, comme existe le monde des réalités physiques; l’un et l’autre indépendants de nous, tous deux de création divine."

“如果我没有弄错的话，有一个完全由数学真理组成的世界，我们只能通过智力接触到这个世界，就像存在一个物理现实的世界一样；两者都独立于我们，并且都是神圣创造的。”

这个概念对于本次讲座所考虑的应用来说，相当于“整数的可计算函数”（即，定义使得能够实际计算对于每个整数的值）的概念。要讨论的过程并不是作用于整数，而是作用于公式，但由于可以枚举相关公式，它们总是可以还原为作用于整数的过程。

此处及后文提到的“数学”一词，始终被理解为“纯数学”（这当然包括从特定立场看被认为正确的形式逻辑）。

在对物理几何学等非数学学科的公理化中，纯数学是预设的；而公理化只涉及超出纯数学部分的学科内容。至少在迄今遇到的例子中，这些内容可以用有限数量的公理来表达。

这种情况在通常的公理表述中并不直接显现，但通过对公理含义的更深入分析可以看出。

集合”操作本质上与“幂集”操作相同，其中集合的幂集被定义为的所有子集的集合。

为了执行迭代，可以令，并假设任何集合都被指定了一个特殊的良序关系。对于第二类序数（极限序数），总是构造之前所得集合的集合。

如果假设直觉主义或有限主义立场，这个定理需要添加一个假设，即集合论公理的一致性，这在集合论被认为属于纯数学的情况下是显然的（因此可以作为假设省略）。然而，对于有限主义数学，一个类似的定理可以在不涉及任何可疑一致性假设的情况下成立；具体来说，引入更高阶的递归函数可以解决越来越多的特定类型的数论问题。在直觉主义数学中，通过引入（通过新公理）第二数类的更大序数，毫无疑问也会有类似的定理成立。

这个假设可以被一致性假设所取代（如罗瑟在其 1936 年的论文中所示），但不可判定命题的结构会变得稍微复杂一些。此外，必须添加假设，即这些公理蕴涵加法、乘法和“

它是不可判定的命题之一，前提是没有错误的数论命题可推导（参见前述定理）。

即皮亚诺公理和通过普通归纳定义的规则，加上满足最严格有限主义要求的逻辑。

如果他只是说“我相信我将能够一个接一个地感知它们是真实的”（假定它们的数量是无限的），他就没有自相矛盾。（见下文。）

对于这一点（或者关于公理一致性的推论）将构成一种不能从所考虑的公理和规则中推导出的数学洞见，这与假设相悖。

例如，可以想象（尽管远超出现代科学的范围），脑生理学可能发展到以下程度，并通过经验得出确凿的结论：

大脑足以解释所有心理现象，并且在图灵意义上是一个机器；
精确地知道负责数学思维的大脑部分的解剖结构和生理功能。

此外，如果采用有限主义（或直觉主义）立场，这种归纳推断可能基于一种（或多或少是经验的）信念，即非有限主义（或非直觉主义）数学是一致的。

当然，思维机制的物理工作可能完全可以被理解；然而，对该特定机制必须总是得出正确（或仅仅一致）结果的洞察，超出了人类理性的能力。对于直觉主义者和有限主义者来说，该定理作为一个蕴涵（而不是析取）成立。需要注意的是，直觉主义者总是断言析取的第一项（并否定第二项，意思是不可判定的命题不能存在）。[见上文，第[?]页。] 但这对于直觉主义数学中的哪种选择适用毫无意义，如果出现在其中的术语被理解为直觉主义者拒绝为无意义的客观意义。

我们无法在文本中找到戈德尔可能在此引用的具体位置。

K. 门格尔的“蕴涵主义立场”（见门格尔 1930a，第 323 页）如果被严格执行，将导致这样的态度，因为根据它，唯一有意义的数学命题（即，用我的术语，唯一属于纯数学的命题）是关于从某些公理和推理规则得出某些结论的命题。

这里得出的结论没有足够普适性的术语能够准确表达，其仅说明，数学的对象和定理与物理世界一样，是客观且独立于我们的自由选择和创造行为的。然而，这并未确定这些客观实体究竟是什么——尤其没有确定它们是位于自然界、人类心智之中，还是两者之外。这三种关于数学本质的观点恰好对应了三种关于概念本质的传统观点，即心理主义、亚里士多德概念论和柏拉图主义。

也就是说，数学对象以及我们认识它们的方式本质上与物理或心理对象及自然规律没有什么不同。然而，真实的情况恰恰相反：如果假设数学的客观性，那么立刻可以得出结论，其对象必须与感性对象完全不同，因为：

如果适当地分析数学命题，会发现它们并未断言任何关于时空世界实际状态的内容。特别是在应用命题中，这一点尤为清楚，例如：“昨天要么下雨，要么没有下雨。”这一评论并不排除满足这些要求的纯概念知识（包括数学之外的知识）的存在。
数学对象是精确地被认识的，普遍规律可以被确知，即通过演绎推理而非归纳推理。
它们可以（原则上）无需依赖感官而被认识（即仅通过理性认识），原因正是它们并不涉及感官（包括内在感官）向我们传递的实际状态，而是涉及可能性和不可能性。

对两个不太复杂或不太人工构造的数论函数之间相等性（而非不等性）的验证，确实能够极大地证明它们完全相等的可能性，尽管在当前科学水平下无法估算其数值。然而，举一些整数上的一般命题为例，其概率现在已经可以估算。例如，关于命题“对于每个，在的十进制展开中从第位到第位之间至少有一个数字不等于 0 的概率”，随着越来越大而趋近于 1。类似的情形也适用于哥德巴赫猜想和费马大定理。

更确切地说，这表明数学的状况与自然科学的状况并没有很大不同。然而，究竟先验主义还是经验主义在最终分析中是正确的，这又是另一个问题。

也就是说，每个问题都必须可以归约为某种有限计算。

声称这些限制是由于一致性要求（这是我们自由选择的结果）而产生的，这种说法毫无意义，因为人们可能选择带来一致性和某些定理。说这些定理仅重复（完全或部分地）最初发明的性质也无助于问题的解决，因为假设最初设定的内容完全被实现，那么它必须足够决定理论中的任何问题，而这被之前的第一个论据和后面的第三个论据所反驳。关于未决命题是否可以通过新的创造行为随意决定的问题，见注释 [？]。

手稿中没有脚注处理这个问题。然而，第29页的速记注释（见下方编辑者注 g 和文本注释）确实包含短语“连续创造”（“continuous creation”）。这可能是戈德尔写给自己关于这个问题的写作笔记。

换句话说，这些约定不能涉及任何语言外的对象（如“证明-定义”那样），而必须仅仅基于符号表达的外在结构，规定其意义或真值的规则。此外，当然这些规则必须是这样的，它们不能暗含任何事实命题的真或假（因为如果这样，它们显然不能被称为无内容或句法的）。然而，这一点意味着这些规则必须是一致的，因为在此讨论的经典逻辑中，不一致会导致所有事实命题都被暗含为真。需要注意的是，如果“句法规则”一词以如此广泛的意义被理解，那么正在讨论的观点包括数学的形式主义基础作为其一个特别的扩展形式。根据形式主义，数学完全基于某些句法规则，例如：具有某种结构的命题被视为真（即公理）；如果某种结构的命题为真，那么另一些命题也为真。此外，很容易看出，一致性证明提供了保证，这些规则是无内容的，因为它们不暗含任何事实命题。另一方面，反过来也成立。下面将显示，唯名论计划的可行性暗示了形式主义计划的可行性。（有关这一唯名论观点的哲学方面的非常清晰的阐述，请参见 Hahn 1935 或 Carnap 1935a, 1955b。）有人可能会怀疑，是否应该将这种唯名论观点归入认为数学是心灵自由创造的观点之下，因为它完全否认数学对象的存在。然而，这两者之间的关系极其密切，因为在另一种观点下，所谓的数学对象的“存在”仅仅在于它们在思想中被建构，而唯名论者不会否认我们实际上想象（不存在的）对象在数学符号背后，并且这些主观的想法甚至可能提供选择句法规则的指导原则。

双竖线表示手稿中标记为“从这里到第29页省略”的内容。由于这些内容没有被划掉，一个合理的推测是，这些内容可能仅仅是要从他的口头演讲中省略。然而，也可能存在其他推测，例如，他后来认为这些内容与1953/9中的讨论重复或被取代。详见文字注释。

哥德尔在其他地方写作“demonstrat.-def.”时没有连字符。它也可能被解读为“演示定义”（这在风格上可能更有吸引力）。

哥德尔的意图可以通过以下段落得知：

当然需要注意，“证明-定义”并不意味着用手指指向为其引入名称的对象（大多数情况下，即使是物理概念也不可能这样做），而是通过解释一个词语在使用情境中的意义来说明其含义。

（摘自哥德尔的第58号脚注。在从第29页打印的替代文本版本中，这一注释被标记为脚注 [详见文字注释]，并且也在哥德尔的第26号脚注中被引用，但我们未在文本中找到任何参考。）

双竖线表示手稿中标记为“从这里到第29页省略”的内容。由于这些内容没有被划掉，一个合理的推测是，这些内容可能仅仅是要从他的口头演讲中省略。然而，也可能存在其他推测，例如，他后来认为这些内容与1959/9中的讨论重复或被取代。详见文字注释。

关于一致性要求，详见脚注[23?]。

也有可能哥德尔打算在这一主题上写一篇新注释。在手稿中，我们给出的文本位于一些被划掉的文字之上，这些文字中提到了一些关于“一致性要求”的内容，但哥德尔可能认为这些内容重复了（我们认为的）第23号注释中的观点。

参见 Ramsey 1926，第368页和382页，及 Carnap 1937，第39页和110页。值得一提的是，Ramsey 甚至通过显式定义成功地将这些内容简化为显式重言式（详见[第24页?]），但代价是承认无限（甚至是超限）长度的命题，这当然需要假设超限集合论以处理这些无限实体。Carnap 将自己限制于有限长度的命题，但必须考虑这些有限命题的无限集、集合的集合等。

例如，对于集合论的任何公理系统，包括选择公理，它可以通过更高阶的公理（或通过的一致性公理）证明一致性，而不需要选择公理。同样，也不是不可能，通过更高阶公理证明层次中较低水平的公理的一致性，虽然会有一些限制，但这些限制可以使直觉主义者接受。

此类抽象概念的例子包括“集合”、“整数函数”、“可证明”（非形式主义意义上的“可被认知为真”）、“可推导”等，或者最终的“存在”，指的是所有可能的符号组合。经典数学一致性证明对这些概念的必要性源于符号可以映射到整数，而因此有限论（更不用说经典）数论包含了所有仅基于它们的证明。到目前为止，这一事实的证据尚未完全确凿，因为与所讨论的非抽象概念有关的显然的公理尚未得到充分研究。然而，即使是领先的形式主义者也承认这一事实；参见 [Bernays 1941a，第144页和147页；1935，第68页和69页；1935b，第94页；1954，第2页；另见 Centzen 1937，第203页]。

此处针对数论句法基础提出的反对意见实质上与 Poincaré 针对 Frege 和 Hilbert 数论基础的反对意见相同。然而，这一反对意见对 Frege 并不成立，因为 Frege 必须预设的逻辑概念和公理并未显式包含“有限多样体”及其公理的概念，而设置句法规则和确立其重言性质所需的语法概念和考量则包含了这一点。

哥德尔手稿第29页底部引用了此处未编号的注释。它既不是正式的脚注，也不是正文插入，而是一条简短的注释。其内容的抄录与翻译见文本注释部分。

这一点同样适用于可以简化为句法规则的数学部分（见上文）。这些规则基于“有限多样体”的概念（即有限符号序列），而这一概念及其性质完全独立于我们的自由选择。事实上，其理论等同于整数理论。构造这样一种语言，将这一理论以句法规则的形式纳入其中的可能性并不能证明任何东西。详见脚注 [?]。

对于哥德尔可能指涉的内容的猜测是，他指的是第35号脚注，但文本中并未提及该脚注。内容如下：

更准确地说，与被批评的观点相对的真实情况如下：

数学术语的意义并不能简化为关于其使用的语言规则，除非是在数学的非常有限领域内（参见 [第25–27页?]）。
即使在这种简化可能的情况下，这些语言规则也不能被视为人为的，关于它们的命题也不能被认为缺乏客观内容，因为这些规则基于“有限多样体”（以有限符号序列的形式）的概念，而这一概念（及其所有性质）完全独立于任何约定和自由选择（因此是客观的）。事实上，其理论等同于算术。

然而，这条注释可能已经被我们的第29号注释（即哥德尔的第49号注释）所取代。

在哥德尔的手稿中，这一页的剩余部分和下一页的一部分内容被划掉。

参见 Darboux 1912[, 第142页]。所引段落如下：

“这些仅因我们心智的弱点而显得不同；但对于一个更强大的智力来说，它们是一回事，其统一性部分地展现在抽象数学与天文学以及物理学所有分支之间的这种奇妙对应中。”

在此，Hermite 似乎转向了亚里士多德的实在论。然而，他的这种转向只是象征性的，因为柏拉图主义仍然是人类心智所能理解的唯一概念。

Some basic theorems on the foundations of mathematics and their implications (1951)

Kurt Gödel

Research in the foundations of mathematics during the past few decades has produced some results which seem to me of interest, not only in themselves, but also with regard to their implications for the traditional philosophical problems about the nature of mathematics. The results themselves, I believe, are fairly widely known, but nevertheless I think it will be useful to present them in outline once again, especially in view of the fact that, due to the work of various mathematicians, they have taken on a much more satisfactory form than they had had originally. The greatest improvement was made possible through the precise definition of the concept of finite procedure, which plays a decisive role in these results. There are several different ways of arriving at such a definition, which, however, all lead to exactly the same concept. The most satisfactory way, in my opinion, is that of reducing the concept of finite procedure to that of a machine with a finite number of parts, as has been done by the British mathematician Turing. As to the philosophical consequences of the results under consideration, I don’t think they have ever been adequately discussed, or [have] only [just been] taken notice of.

The metamathematical results I have in mind are all centered around, or, one may even say, are only different aspects of one basic fact, which might be called the incompletability or inexhaustibility of mathematics. This fact is encountered in its simplest form when the axiomatic method is applied, not to some hypothetico-deductive system such as geometry (where the mathematician can assert only the conditional truth of the theorems), but to mathematics proper, that is, to the body of those mathematical propositions which hold in an absolute sense, without any further hypothesis. There must exist propositions of this kind, because otherwise there could not exist any hypothetical theorems either. For example, some implications of the form:

If such and such axioms are assumed, then such and such a theorem holds,

must necessarily be true in an absolute sense. Similarly, any theorem of finitistic number theory, such as 2 + 2 = 4, is, no doubt, of this kind. Of course, the task of axiomatizing mathematics proper differs from the usual conception of axiomatics insofar as the axioms are not arbitrary, but must be correct mathematical propositions, and moreover, evident without proof. There is no escaping the necessity of assuming some axioms or rules of inference as evident without proof, because the proofs must have some starting point. However, there are widely divergent views as to the extension of mathematics proper, as I defined it. The intuitionists and finitists, for example, reject some of its axioms and concepts, which others acknowledge, such as the law of excluded middle or the general concept of set.

The phenomenon of the inexhaustibility of mathematics, however, always is present in some form, no matter what standpoint is taken. So I might as well explain it for the simplest and most natural standpoint, which takes mathematics as it is, without curtailing it by any criticism. From this standpoint all of mathematics is reducible to abstract set theory. For example, the statement that the axioms of projective geometry imply a certain theorem means that if a set ( M ) of elements called points and a set ( N ) of subsets of ( M ) called straight lines satisfy the axioms, then the theorem holds for ( N, M ). Or, to mention another example, a theorem of number theory can be interpreted to be an assertion about finite sets.

So the problem at stake is that of axiomatizing set theory. Now, if one attacks this problem, the result is quite different from what one would have expected. Instead of ending up with a finite number of axioms, as in geometry, one is faced with an infinite series of axioms, which can be extended further and further, without any end being visible and, apparently, without any possibility of comprising all these axioms in a finite rule producing them.

This comes about through the circumstance that, if one wants to avoid the paradoxes of set theory without bringing in something entirely extraneous to actual mathematical procedure, the concept of set must be axiomatized in a stepwise manner. If, for example, we begin with the integers, that is, the finite sets of a special kind, we have at first the sets of integers and the axioms referring to them (axioms of the first level), then the sets of sets of integers with their axioms (axioms of the second level), and so on for any finite iteration of the operation “set of”. Next we have the set of all these sets of finite order. But now we can deal with this set in exactly the same manner as we dealt with the set of integers before, that is, consider the subsets of it (that is, the sets of order ( \omega )) and formulate axioms about their existence.

Evidently this procedure can be iterated beyond ( \omega ), in fact up to any transfinite ordinal number. So it may be required as the next axiom that the iteration is possible for any ordinal, that is, for any order type belonging to some well-ordered set. But are we at an end now? By no means. For we have now a new operation of forming sets, namely, forming a set out of some initial set ( A ) and some well-ordered set ( B ) by applying the operation “set of” to ( A ) as many times as the well-ordered set ( B ) indicates. And, setting ( B ) equal to some well-ordering of ( A ), now we can iterate this new operation, and again iterate it into the transfinite. This will give rise to a new operation again, which we can treat in the same way, and so on. So the next step will be to require that any operation producing sets out of sets can be iterated up to ( | ) any ordinal number (that is, order type of a well-ordered set). But are we at an end now? No, because we can require not only that the procedure just described can be carried out with any
operation, but that moreover there should exist a set closed with respect
to it, that is, one which has the property that, if this procedure (with any
operation) is applied to elements of this set, it again yields elements of this
set. You will realize, I think, that we are still not at an end, nor can there
ever be an end to this procedure of forming the axioms, because the very
formulation of the axioms up to a certain stage gives rise to the next axiom.
It is true that in the mathematics of today the higher levels of this hierar-
chy are practically never used. It is safe to say that 99.9% of present-day
mathematics is contained in the first three levels of this hierarchy. So for all
practical purposes, all of mathematics can be reduced to a finite number of
axioms. However, this is a mere historical accident, which is of no impor-
tance for questions of principle. Moreover it is not altogether unlikely that
this character of present-day mathematics may have something to do with
another character of it, namely, its inability to prove certain fundamental
theorems, such as, for example, Riemann’s hypothesis, in spite of many
years of effort. For it can be shown that the axioms for sets of high levels,
in their relevance, are by no means confined to these sets, but, on the con-
trary, have consequences even for the 0-level, that is, the theory of integers.
To be more exact, each of these set-theoretical axioms entails the solution
of certain diophantine problems which had been undecidable on the basis
of the preceding axioms.
The diophantine problems in question are of the
following type: Let ( P(x₁, ..., xₙ, y₁, ..., yₘ) ) be a polynomial with given in-
tegral coefficients and ( n + m ) variables, ({x₁, ..., xₙ, y₁, ..., yₘ}), and consider
the variables (x₁) as the unknowns and the variables (y₁) as parameters; then
the problem is: Has the equation (P = 0) integral solutions for any integral
values of the parameters, or are there integral values of the parameters for
which this equation has no integral solutions? To each of the set-theoretical
axioms a certain polynomial (P) can be assigned, for which the problem just
formulated becomes decidable owing to this axiom. It even can always be
achieved that the degree of (P) is not higher than 4. [The] mathematics of
today has not yet learned to make use of the set-theoretical axioms for
the solution of number-theoretical problems, except for the axioms of the
first level. These are actually used in analytic number theory. But for
mastering number theory this is demonstrably insufficient. Some kind of set-theoretical number theory, still to be discovered, would certainly reach much farther.

I have tried so far to explain the fact I call [the] incompletability of mathematics for one particular approach to the foundations of mathematics, namely axiomatics of set theory. That, however, this fact is entirely independent of the particular approach and standpoint chosen appears from certain very general theorems. The first of these theorems simply states that, whatever well-defined system of axioms and rules of inference may be chosen, there always exist diophantine problems of the type described which are undecidable by these axioms and rules, provided only that no false propositions of this type are derivable.
If I speak of a well-defined system of axioms and rules here, this only means that it must be possible actually to write the axioms down in some precise formalism or, if their number is infinite, a finite procedure for writing them down one after the other must be given. Likewise the rules of inference are to be such that, given any premises, either the conclusion (by any one of the rules of inference) can be written down, or it can be ascertained that there exists no immediate conclusion by the rule of inference under consideration. This requirement for the rules and axioms is equivalent to the requirement that it should be possible to build a finite machine, in the precise sense of a "Turing machine", which will write down all the consequences of the axioms one after the other. For this reason, the theorem under consideration is equivalent to the fact that there exists no finite procedure for the systematic decision of all diophantine problems of the type specified.

The second theorem has to do with the concept of freedom from contradiction. For a well-defined system of axioms and rules the question of their consistency is, of course, itself a well-defined mathematical question. Moreover, since the symbols and propositions of [any] one formalism are always at most enumerable, everything can be mapped onto the integers, and it is plausible and in fact demonstrable that the question of consistency can always be transformed into a number-theoretical question (to be more exact, into one of the type described above). Now the theorem says that **for any well-defined system of axioms and rules, in particular, the proposition stating their consistency (or rather the equivalent number-theoretical proposition) is undemonstrable from these axioms and rules, provided these axioms and rules are consistent and suffice to derive a certain portion of the finitistic arithmetic of integers. It is this theorem which makes the incompletability of mathematics particularly evident. For, it makes it impossible that someone should set up a certain well-defined system of axioms and rules and consistently make the following assertion about it: All of these axioms and rules I perceive (with mathematical certitude) to be correct, and moreover I believe that they contain all of mathematics. If someone makes such a statement he contradicts himself. (^{11}) For if he perceives the axioms under consideration to be correct, he also perceives (with the same certainty) that they are consistent. Hence he has a mathematical insight not derivable from his axioms. However, one has to be careful in order to understand clearly the meaning of this state of affairs. Does it mean that no well-defined system of correct axioms can contain all of mathematics proper? It does, if by mathematics proper is understood the system of all true mathematical propositions; it does not, however, if one understands by it the system of all demonstrable mathematical propositions. I shall distinguish these two meanings of mathematics as mathematics in the objective and in the subjective sense: Evidently no well-defined system of correct axioms can comprise all [of] objective mathematics, since the proposition which states the consistency of the system is true, but not demonstrable in the system. However, as to subjective mathematics, it is not precluded that there should exist a finite rule producing all its evident axioms. However, if such a rule exists, we with our human understanding could certainly never know it to be such, that is, we could never know with mathematical certanity that all propositions it produces are correct; (^{12}) or in other terms, we could perceive to be true only one proposition after the other, for any finite number of them. The assertion, however, that they are all true could at most be known with empirical certainty, on the basis of a sufficient number of instances or by other inductive inferences. (^{13}) If it were so, this would mean that the human mind (in the realm of pure mathematics) is equivalent to a finite machine that, however, is unable to understand completely (^{14}) its own functioning. This inability [of man] to understand himself would then wrongly appear to him as its [the mind's] boundlessness or inexhaustibility. But, please, note that if it were so, this would in no way derogate from the incompletability of objective mathematics. On the contrary, it would only make it particularly striking. For if the human mind were equivalent to a finite machine, then objective mathematics not only would be incompletable in the sense of not being contained in any well-defined axiomatic system, but moreover there would exist absolutely unsolvable diophantine problems of the type described above, where the epithet “absolutely” means that they would be undecidable, not just within some particular axiomatic system, but by any mathematical proof the human mind can conceive. So the following disjunctive conclusion is inevitable: Either mathematics is incompletable in this sense, that its evident axioms can never be comprised in a finite rule, that is to say, the human mind (even within the realm of pure mathematics) infinitely surpasses the powers of any finite machine, or else there exist absolutely unsolvable diophantine problems of the type specified (where the case that both terms of the disjunction are true is not excluded, so that there are, strictly speaking, three alternatives). It is this mathematically established fact which seems to me of great philosophical interest. Of course, in this connection it is of great importance that at least this fact is entirely independent of the special standpoint taken toward the foundations of mathematics. (^{15})

There is, however, one restriction to this independence, namely, the standpoint taken must be liberal enough to admit propositions about all integers as meaningful. If someone were so strict a finitist that he would maintain that only particular propositions of the type (2 + 2 = 4) belong to mathematics proper, (^{16}) then the incompletability theorem would not apply—at least not this incompletability theorem. But I don’t think that such an attitude could be maintained consistently, because it is by exactly the same kind of evidence that we judge that (2+2=4) and that (a+b=b+a) for any two integers (a, b). Moreover, this standpoint, in order to be consistent, would have to exclude also concepts that refer to all integers, such as “+” (or to all formulas, such as “correct proof by such and such rules”) and replace them with others that apply only within some finite domain of integers (or formulas). It is to be noted, however, that although the truth of the disjunctive theorem is independent of the standpoint taken, the question as to which alternative holds need not be independent of it. (See footnote [15].)

I think I now have explained sufficiently the mathematical aspect of the situation and can turn to the philosophical implications. Of course, in consequence of the undeveloped state of philosophy in our days, you must not expect these inferences to be drawn with mathematical rigour.

Corresponding to the disjunctive form of the main theorem about the incompletability of mathematics, the philosophical implications prima facie will be disjunctive too; however, under either alternative they are very decidedly opposed to materialistic philosophy. Namely, if the first alternative holds, this seems to imply that the working of the human mind cannot be reduced to the working of the brain, which to all appearances is a finite machine with a finite number of parts, namely, the neurons and their connections. So apparently one is driven to take some vitalistic viewpoint. On the other hand, the second alternative, where there exist absolutely undecidable mathematical propositions, seems to disprove the view that mathematics is only our own creation; for the creator necessarily knows all properties of his creatures, because they can’t have any others except those he has given to them. So this alternative seems to imply that mathematical objects and facts (or at least something in them) exist objectively and independently of our mental acts and decisions, that is to say, [it seems to imply] some form or other of Platonism or “realism” as to the mathematical objects.For, the empirical interpretation of mathematics, that is, the view that mathematical facts are a special kind of physical or psychological facts, is too absurd to be seriously maintained (see below). It is not known whether the first alternative holds, but at any rate it is in good agreement with the opinions of some of the leading men in brain and nerve physiology, who very decidedly deny the possibility of a purely mechanistic explanation of psychical and nervous processes.

As far as the second alternative is concerned, one might object that the constructor need not necessarily know every property of what he constructs. For example, we build machines and still cannot predict their behaviour in every detail. But this objection is very poor. For we don’t create the machines out of nothing, but build them out of some given material. If the situation were similar in mathematics, then this material or basis for our constructions would be something objective and would force some realistic viewpoint upon us even if certain other ingredients of mathematics were our own creation. The same would be true if in our creations we were to use some instrument in us but different from our ego (such as “reason” interpreted as something like a thinking machine). For mathematical facts would then (at least in part) express properties of this instrument, which would have an objective existence.

One may thirdly object that the meaning of a proposition about all integers, since it is impossible to verify it for all integers one by one, can consist only in the existence of a general proof. Therefore, in the case of an undecidable proposition about all integers, neither itself nor its negation is true. Hence neither expresses an objectively existing but unknown property of the integers. I am not in a position now to discuss the epistemological question as to whether this opinion is at all consistent. It certainly looks as if one must first understand the meaning of a proposition before he can understand a proof of it, so that the meaning of "all" could not be defined in terms of the meaning of "proof". But independently of this epistemological investigation, I wish to point out that one may conjecture the truth of a universal proposition (for example, that I shall be able to verify a certain property for any integer given to me) and at the same time conjecture that no general proof for this fact exists. It is easy to imagine situations in which both these conjectures would be very well founded. For the first half of it, this would, for example, be the case if the proposition in question were some equation (F(n) = G(n)) of two number-theoretical functions which could be verified up to very great numbers (n). Moreover, exactly as in the natural sciences, this inductio per enumerationem simplicem is by no means the only inductive method conceivable in mathematics. I admit that every mathematician has an inborn abhorrence to giving more than heuristic significance to such inductive arguments. I think, however, that this is due to the very prejudice that mathematical objects somehow have no real existence. If mathematics describes an objective world just like physics, there is no reason why inductive methods should not be applied in mathematics just the same as in physics. The fact is that in mathematics we still have the same attitude today that in former times one had toward all science, namely, we try to derive everything by cogent proofs from the definitions (that is, in ontological terminology, from the essences of things). Perhaps this method, if it claims monopoly, is as wrong in mathematics as it was in physics.

This whole consideration incidentally shows that the philosophical implications of the mathematical facts explained do not lie entirely on the side of rationalistic or idealistic philosophy, but that in one respect they favor the empiricist viewpoint. It is true that only the second alternative points in this direction.Howerve,and this is the item i would to discuss now, it seems to me that the philosophical conclusions drawn under the second alternative, in particular, conceptual realism (Platonism), are supported by modern developments in the foundations of mathematics also, irrespectively of which alternative holds. The main arguments pointing in this direction seem to me [to be] the following. First of all, if mathematics were our free creation, ignorance as to the objects we created, it is true, might still occur, but only through lack of a clear realization as to what we really have created (or, perhaps, due to the practical difficulty of too complicated computations). Therefore it would have to disappear (at least in principle, although perhaps not in practice (^{21})) as soon as we attain perfect clearness. However, modern developments in the foundations of mathematics have accomplished an insurmountable degree of exactness, but this has helped practically nothing for the solution of mathematical problems.

Secondly, the activity of the mathematician shows very little of the freedom a creator should enjoy. Even if, for example, the axioms about integers were a free invention, still it must be admitted that the mathematician, after he has imagined the first few properties of his objects, is at an end with his creative ability, and he is not in a position also to create the validity of the theorems at his will. If anything like creation exists at all in mathematics, then what any theorem does is exactly to restrict the freedom of creation. That, however, which restricts it must evidently exist independently of the creation.(^{22})

Thirdly, if mathematical objects are our creations, then evidently integers and sets of integers will have to be two different creations, the first of which does not necessitate the second. However, in order to prove certain propositions about integers, the concept of set of integers is necessary. So here, in order to find out what properties we have [given] to certain objects of our imagination, [we] must first create certain other objects—a very strange situation indeed!

What I [have] said so far has been formulated in terms of the rather vague concept of "free creation" or "free invention". There exist attempts to give a more precise meaning to this term. However, this only has the consequence that also the disproof of the standpoint in question is becoming more precise and cogent. I would like to show this in detail for the most precise, and at the same time most radical, formulation that has been given so far. It is that which ( c | ) interprets mathematical propositions as expressing solely certain aspects of syntactical (or linguistic)( ^{23} ) conventions, that is,they simply repeat parts of these conventions. According to this view, mathematical propositions, duly analyzed, must turn out to be as void of content as, for example, the statement "All stallions are horses". Everybody will agree that this proposition does not express any zoological or other objective fact, but [rather,] its truth is due solely to the circumstance that we chose to use the term "stallion" as an abbreviation for "male horse".

Now by far the most common type of symbolic conventions are definitions (either explicit or contextual, where the latter however must be such as to make it possible to eliminate the term defined in any context [where] it occurs). Therefore the simplest version of the view in question would consist in the assertion that mathematical propositions are true solely owing to the definitions of the terms occurring in them, that is, that by successively replacing all terms by their definition, any theorem can be reduced to an explicit tautology, ( a = a ). (Note that ( a = a ) must be admitted as true if definitions are admitted, for one may define ( b ) by ( b = a ) and then, owing to this definition, replace ( b ) by ( a ) in this equality.) But now it follows directly from the theorems mentioned before that such a reduction to explicit tautologies is impossible. For it would immediately yield a mechanical procedure for deciding about the truth or falsehood of every mathematical proposition. Such a procedure, however, cannot exist, not even for number theory.

This disproof, it is true, refers only to the simplest version of this (nominalistic) standpoint. But the more refined ones do not fare any better. The weakest statement that at least would have to be demonstrable, in order that this view concerning the tautological character of mathematics be tenable, is the following: Every demonstrable mathematical proposition can be deduced from the rules about the truth and falsehood of sentences alone (that is, without using or knowing anything else except these rules) and the negations of demonstrable mathematical propositions cannot be so derived.

In precisely formulated languages, such rules (that is, rules which stipulate under which conditions a given sentence is true) occur as a means for determining the meaning of sentences. Moreover in all known languages there are propositions which seem to be true owing to these rules alone. For example, if disjunction and negation are introduced by those rules:

( p \lor q ) is true if at least one of its terms is true, and
( \sim p ) is true if ( p ) is not true.

Then it clearly follows from these rules that ( p \lor \sim p ) is always true whatever ( p ) may be. (Propositions so derivable are called tautologies.) Now it is actually so, that for the symbolisms of mathematical logic, with suitably chosen semantical rules, the truth of the mathematical axioms is derivable from these rules;^25 however (and this is the great stumbling block), in this derivation the mathematical and logical concepts and axioms themselves must be used in a special application, namely, as referring to symbols, combinations of symbols, sets of such combinations, etc. Hence this theory, if it wants to prove the tautological character of the mathematical axioms, must first assume these axioms to be true. So while the original idea of this viewpoint was to make the truth of the mathematical axioms understandable by showing that they are tautologies, it ends up with just the opposite, that is, the truth of the axioms must first be assumed and then it can be shown that, in a suitably chosen language, they are tautologies.

Moreover, a similar statement holds good for the mathematical concepts, that is, instead of being able to define their meaning by means of symbolic conventions, one must first know their meaning in order to understand the syntactical conventions in question or the proof that they imply the mathematical axioms but not their negations. Now, of course, it is clear that this elaboration of the nominalistic view does not satisfy the requirement set up on page [25?], because not the syntactic rules alone, but all of mathematics in addition is used in the derivations. But moreover, this elaboration of nominalism would yield an outright disproof of it (I must confess I can’t picture any better disproof of this view than this proof of it), provided that one thing could be added, namely, that the outcome described is unavoidable (that is, independent of the particular symbolic language and interpretation of mathematics chosen). Now it is not exactly this that can be proved, but something so close to it that it also suffices to disprove the view in question. Namely, it follows by the metatheorems mentioned that a proof for the tautological character (in a suitable language) of the mathematical axioms is at the same time a proof for their consistency, and cannot be achieved with any weaker means of proof than are contained in these axioms themselves. This does not mean that all the axioms of a given system must be used in its consistency proof. On the contrary, usually the axioms lying outside the system which are necessary make it possible to dispense with some of the axioms of the system (although they do not imply the latter).^26 However, what follows with practical certainty is this: In order to prove the consistency of classical number theory (and a fortiori of all stronger systems) certain abstract concepts (and the directly evident axioms referring to them) must be used, where “abstract” means concepts which do not refer to sense objects,^27 of which symbols are a special kind. These abstract concepts, however, are certainly not syntactical [but rather those whose justification by syntactical considerations should be the main task of nominalism]. Hence it follows that there exists no rational justification of our precritical beliefs concerning the applicability and consistency of classical mathematics (nor even its undermost level, number theory) on the basis of a syntactical interpretation. It is true that this statement does not apply to certain subsystems of classical mathematics, which may even contain some part of the theory of the abstract concepts referred to. In this sense, nominalism can point to some partial successes. For it is actually possible to base the axioms of these systems on purely syntactical considerations. In this manner, for example, the use of the concepts of “all” and “there is” referring to integers can be justified (that is, proved consistent) by means of syntactical considerations. However, for the most essential number-theoretic axiom, complete induction, such a syntactical foundation, even within the limits in which it is possible, gives no justification of our precritical belief in it, since this axiom itself has to be used.in the syntactical considerations.^28 The fact that the more modest you are in the axioms for which you want to set up a tautological interpretation, the less of mathematics you need in order to do it, has the consequence that if finally you become so modest as to confine yourself to some finite domain, for example, to the integers up to 1000, then the mathematical propositions valid in this field can be so interpreted as to be tautological even in the strictest sense, that is, reducible to explicit tautologies by means of the explicit definitions of the terms. No wonder, because the section of mathematics necessary for the proof of the consistency of this finite mathematics is contained already in the theory of the finite combinatorial processes which are necessary in order to reduce a formula to an explicit tautology by substitutions. This explains the well-known, but misleading, fact that formulas like ( 5 + 7 = 12 ) can, by means of certain definitions, be reduced to explicit tautologies. This fact, incidentally, is misleading also for the reason that in these reductions (if they are to be interpreted as simple substitutions of the definiens for the definiendum on the basis of explicit definitions), the ( + ) is not identical with the ordinary ( + ), because it can be defined only for a finite number of arguments (by an enumeration of this finite number of cases). If, on the other hand, ( + ) is defined contextually, then one has to use the concept of finite manifold already in the proof of ( 2 + 2 = 4 ). A similar circularity also occurs in the proof that ( p \lor \neg p ) is a tautology, because disjunction and negation, in their intuitive meanings, evidently occur in it.

The essence of this view is that there exists no such thing as a mathematical fact, that the truth of propositions which we believe express mathematical facts only means that (due to the rather complicated rules which define the meaning of propositions, that is, which determine under what circumstances a given proposition is true) an idle running of language occurs in these propositions, in that the said rules make them true no matter what the facts are. Such propositions can rightly be called void of content. Now it [is] actually possible to build up a language in which mathematical propositions are void of content in this sense. The only trouble is:

that one has to use the very same mathematical facts^8 (or equally complicated other mathematical facts) in order to show that they don’t exist;
that by this method, if a division of the empirical facts into two parts, ( A ) and ( B ), is given such that ( B ) implies nothing in ( A ), a language can be constructed in which the propositions expressing ( B ) would be void of content. And if your opponent were to say: "You are arbitrarily disregarding certain observable facts ( B )", one may answer: "You are doing the same thing, for example with the law of complete induction, which I perceive to be true on the basis of my understanding (that is, perception) of the concept of integer."

However it seems to me that nevertheless one ingredient of this wrong theory of mathematical truth is perfectly correct and really discloses the true nature of mathematics. Namely, it is correct that a mathematical proposition says nothing about the physical or psychical reality existing in space and time, because it is true already owing to the meaning of the terms occurring in it, irrespectively of the world of real things. What is wrong, however, is that the meaning of the terms (that is, the concepts they denote) is asserted to be something man-made and consisting merely in semantical conventions. The truth, I believe, is that these concepts form an objective reality of their own, which we cannot create or change, but only perceive and describe.^29

Therefore a mathematical proposition, although it does not say anything about space-time reality, still may have a very sound objective content, insofar as it says something about relations of concepts. The existence of non-"tautological" relations between the concepts of mathematics appears above all in the circumstance that for the primitive terms of mathematics, axioms must be assumed, which are by no means tautologies (in the sense of being in any way reducible to (a = a)), but still do follow from the meaning of the primitive terms under consideration. For example, the basic axiom, or rather, axiom schema, for the concept of set of integers says that, given a well-defined property of integers (that is, a propositional expression (\varphi(n)) with an integer variable (n)), there exists the set (M) of those integers which have the property (\varphi). Now, considering the circumstance that (\varphi) may itself contain the term "set of integers", we have here a series of rather involved axioms about the concept of set. Nevertheless, these axioms (as the aforementioned results show) cannot be reduced to anything substantially simpler, let alone to explicit tautologies. It is true that these axioms are valid owing to the meaning of the term "set"—one might even say they express the very meaning of the term "set"—and therefore they might fittingly be called analytic; however, the term "tautological", that is, devoid of content, for them is entirely out of place, because even the assertion of the existence of a concept of set satisfying these axioms (or of the consistency of these axioms) is so far from being empty that it cannot be proved without again using the concept of set itself, or some other abstract concept of a similar nature.

Of course, this particular argument is addressed only to mathematicians who admit the general concept of set in mathematics proper. For finitists, however, literally the same argument could be alleged for the concept of integer and the axiom of complete induction. For, if the general concept of set is not admitted in mathematics proper, then complete induction must be assumed as an axiom.

I wish to repeat that "analytic" here does not mean "true owing to our definitions", but rather "true owing to the nature of the concepts occurring [therein]", in contradistinction to "true owing to the properties and the behavior of things". This concept of analytic is so far from meaning "void of content" that it is perfectly possible that an analytic proposition might be undecidable (or decidable only with [a certain] probability). For, our knowledge of the world of concepts may be as limited and incomplete as that of [the] world of things. It is certainly undeniable that this knowledge, in certain cases, not only is incomplete, but even indistinct. This occurs in the paradoxes of set theory, which are frequently alleged as a disproof of Platonism, but, I think, quite unjustly. Our visual perceptions sometimes contradict our tactile perceptions, for example, in the case of a rod immersed in water, but nobody in his right mind will conclude from this fact that the outer world does not exist.

I have purposely spoken of two separate worlds (the world of things and the world of concepts), because I do not think that Aristotelian realism (according to which concepts are parts or aspects of things) is tenable.

Of course I do not claim that the foregoing considerations amount to a real proof of this view about the nature of mathematics. The most I could assert would be to have disproved the nominalistic view, which considers mathematics to consist solely in syntactical conventions and their consequences. Moreover, I have adduced some strong arguments against the more general view that mathematics is our own creation. There are, however, other alternatives to Platonism, in particular psychologism and Aristotelian realism. In order to establish Platonistic realism, these theories would have to be disproved one after the other, and then it would have to be shown that they exhaust all possibilities. I am not in a position to do this now; however, I would like to give some indications along these lines. One possible form of psychologism admits that mathematics investigates relations of concepts and that concepts cannot be created at our will, but are given to us as a reality, which we cannot change; however, it contends that these concepts are only psychological dispositions, that is, that they are nothing but, so to speak, the wheels of our thinking machine. To be more exact, a concept would consist in the disposition

to have a certain mental experience when we think of it

and

to pass certain judgments (or have certain experiences of direct knowledge) about its relations to other concepts and to empirical objects.

The essence of this psychologistic view is that the object of mathematics is nothing but the psychological laws by which thoughts, convictions, and so on occur in us, in the same sense as the object of another part of psychology is the laws by which emotions occur in us. The chief objection to this view I can see at the present moment is that if it were correct, we would have no mathematical knowledge whatsoever. We would not know, for example, that 2 + 2 = 4, but only that our mind is so constituted as to hold this to be true, and there would then be no reason whatsoever why, by some other train of thought, we should not arrive at the opposite conclusion with the same degree of certainty. Hence, whoever assumes that there is some domain, however small, of mathematical propositions which we know to be true, cannot accept this view.*

I am under the impression that after sufficient clarification of the concepts in question it will be possible to conduct these discussions with mathematical rigour and that the result then will be that (under certain assumptions which can hardly be denied [in particular the assumption that there exists at all something like mathematical knowledge]) the Platonistic view is the only one tenable. Thereby I mean the view that mathematics describes a non-sensual reality, which exists independently both of the acts and [of] the dispositions of the human mind and is only perceived, and probably perceived very incompletely, by the human mind. This view is rather unpopular among mathematicians; there exist, however, some great mathematicians who have adhered to it. For example, Hermite once wrote the following sentence:

Il existe, si je ne me trompe, tout un monde qui est l’ensemble des vérités mathématiques, dans lequel nous n’avons accès que par l’intelligence, comme existe le monde des réalités physiques; l’un et l’autre indépendants de nous, tous deux de création divine._ 30

[There exists, unless I am mistaken, an entire world consisting of the totality of mathematical truths, which is accessible to us only through our intelligence, just as there exists the world of physical realities; each one is independent of us, both of them divinely created.]

This concept, for the applications to be considered in this lecture, is equivalent to the concept of a "computable function of integers" (that is, one whose definition makes it possible actually to compute f(n) for each integer n). The procedures to be considered do not operate on integers but on formulas, but because of the enumeration of the formulas in question, they can always be reduced to procedures operating on integers.

The term "mathematics", here and in the sequel, is always supposed to mean "mathematics proper" (which of course includes formal logic as far as it is acknowledged to be correct by the particular standpoint taken).

In the axiomatizations of non-mathematical disciplines such as physical geometry, mathematics proper is presupposed; and the axiomatization refers to the content of the discipline under consideration only insofar as it goes beyond mathematics proper. This content, at least in the examples which have been encountered so far, can be expressed by a finite number of axioms.

This circumstance, in the usual presentation of the axioms, is not directly apparent, but shows itself on closer examination of the meaning of the axioms.

The operation "set of" is substantially the same as the operation "power set", where the power set of ( M ) is by definition the set of all subsets of ( M ).

In order to carry out the iteration one may put ( A = B ) and assume that a special well-ordering has been assigned to any set. For ordinals of the second kind [limit ordinals], the set of the previously obtained sets is always to be formed.

This theorem, in order to hold also if the intuitionistic or finitistic standpoint is assumed, requires as a hypothesis the consistency of the axioms of set theory, which of course is self-evident (and therefore can be dropped as a hypothesis) if set theory is considered to be mathematics proper. However, for finitistic mathematics asimilar theorem holds, without any questionable hypothesis of consistency; namely, the introduction of recursive functions of higher and higher order leads to the solution of more and more number-theoretical problems of the specified kind. In intuitionistic mathematics there doubtless holds a similar theorem for the introduction (by new axioms) of greater and greater ordinals of the second number class.

This hypothesis can be replaced by consistency (as shown by Rosser in [his 1936]), but the undecidable propositions then have a slightly more complicated structure. Moreover, the hypothesis must be added that the axioms imply the primitive properties of addition, multiplication and <.>

It is one of the propositions which are undecidable, provided that no false number-theoretical [propositions] are derivable (see the preceding theorem).

Namely, Peano's axioms and the rule of definition by ordinary induction, with a logic satisfying the strictest finitistic requirements.

If he only says "I believe I shall be able to perceive one after the other to be true" (where their number is supposed to be infinite), he does not contradict himself. (See below.)

For this (or the consequence concerning the consistency of the axioms) would constitute a mathematical insight not derivable from the axioms [and] rules under consideration, contrary to the assumption.

For example, it is conceivable (although far outside the limits of present-day science) that brain physiology would advance so far that it would be known with empirical certainty

that the brain suffices for the explanation of all mental phenomena and is a machine in the sense of Turing;
that such and such is the precise anatomical structure and physiological functioning of the part of the brain which performs mathematical thinking.
Furthermore, in case the finitistic (or intuitionistic) standpoint is taken, such an inductive inference might be based on a (more or less empirical) belief that non-finitistic (or non-intuitionistic) mathematics is consistent.

Of course, the physical working of the thinking mechanism could very well be completely understandable; the insight, however, that this particular mechanism must always lead to correct (or only consistent) results would surpass the powers of human reason.

For intuitionists and finitists the theorem holds as an implication (instead of a disjunction). It is to be noted that intuitionists have always asserted the first term of the disjunction (and negated the second term, in the sense that no demonstrably undecidable propositions can exist). [See above, p. [?].] But this means nothing for the question which alternative applies to intuitionistic mathematics, if the terms occurring in it are understood in the objective sense (rejected as meaningless by the intuitionists). As for finitism, it seems very likely that the first disjunctive term is false.

We are unable to locate a place in the text to which Gödel would be referring here.

K. Menger’s “implicationistic standpoint” (see Menger 1930a, p. 323), if taken in the strictest sense, would lead to such an attitude, since according to it, the only meaningful mathematical propositions (that is, in my terminology, the only ones belonging to mathematics proper) would be those that assert that such and such a conclusion can be drawn from such and such axioms and rules of inference in such and such a manner. This, however, is a proposition of exactly the same logical character as (2+2=4). Some of the untenable consequences of this standpoint are the following: A negative proposition to the effect that the conclusion (B) cannot be drawn from the axioms and rules (A) would not belong to mathematics proper; hence nothing could be known about it except perhaps that it follows from certain other axioms and rules. However, a proof that it does so follow (since these other axioms and rules again are arbitrary) would in no way exclude the possibility that (in spite of the formal proof to the contrary) a derivation of (B) from (A) might some day be accomplished. For the same reason also, the usual inductive proof for (a+b=b+a) would not exclude the possibility of discovering two integers not satisfying this equation.

There exists no term of sufficient generality to express exactly the conclusion drawn here, which only says that the objects and theorems of mathematics are as objective and independent of our free choice and our creative acts as is the physical world. It determines, however, in no way what these objective entities are—in particular, whether they are located in nature or in the human mind or in neither of the two. These three views about the nature of mathematics correspond exactly to the three views about the nature of concepts, which traditionally go by the names of psychologism, Aristotelian conceptualism and Platonism.

That is, the view that mathematical objects and the way in which we know them are not essentially different from physical or psychical objects and laws of nature. The true situation, on the contrary, is that if the objectivity of mathematics is assumed, it follows at once that its objects must be totally different from sensual objects because:

Mathematical propositions, if properly analyzed, turn out to assert nothing about the actualities of the space-time world. This is particularly clear in applied propositions such as: Either it has or it has not rained yesterday. The existence of purely conceptual knowledge (besides mathematics) satisfying these requirements is not excluded by this remark.
The mathematical objects are known precisely, and general laws can be recognized with certainty, that is, by deductive, not inductive, inference.
They can be known (in principle) without using the senses (that is, by means of reason alone) for this very reason, that they don’t concern actualities about which the senses (the inner sense included) inform us, but possibilities and impossibilities.

Such a verification of an equality (not an inequality) between two number-theoretical functions of not too complicated or artificial structure would certainly give a great probability to their complete equality, although its numerical value could not be estimated in the present state of science. However, it is easy to give examples of general propositions about integers where the probability can be estimated even now. For example, the probability of the proposition which states that for each (n) there is at least one digit (\neq 0) between the (n)-th and (n^2)-th digits of the decimal expansion of (\pi) converges toward 1 as one goes on verifying it for greater and greater (n). A similar situation also prevails for Goldbach's and Fermat's theorems.

To be more precise, it suggests that the situation in mathematics is not so very different from that in the natural sciences. As to whether, in the last analysis, apriorism or empiricism is correct is a different question.

That is, every problem would have to be reducible to some finite computation.

t is of no avail to say that these restrictions are brought about by the requirement of consistency, which itself is our free choice, because one might choose to bring about consistency and certain theorems. Nor does it help to say that the theorems only repeat (wholly or in part) the properties first invented, because then the exact realization of what was first assumed would have to be sufficient for deciding any question of the theory, which is disproved by the first [argument (above)] and the third argument [(below)]. As to the question of whether undecidable propositions can be decided arbitrarily by a new act of creation, see fn. [?].

No footnote in the manuscript deals with this question. However, the shorthandannotation to p. 29'(see editorial note g below and the textual notes) does contain thephrase“continluouscreation”.That could have been a note of Godel to himself towrite something on the question.

That is, the conventions must not refer to any extralinguistic objects (as does a demonstr[ation]-def[inition]), but must state rules about the meaning or truth of symbolic expressions solely on the basis of their outward structure. Moreover, of course these rules must be such that they do not imply the truth or falsehood of any factual propositions (since in that case they could certainly not be called void of content nor syntactical). This, however, entails their consistency, because an inconsistency (in classical logic, which is under consideration here) would imply every factual proposition. It is to be noted that if the term "syntactical rule" is understood in this generality, the view under consideration includes, as a special elaboration of it, the formalistic foundation of mathematics, since according to the latter, mathematics is based solely on certain syntactical rules of the form: Propositions of such and such structure are true [the axioms], and if propositions of ... structure are true, then such and such other propositions are also true; and moreover, as can easily be seen, the consistency proof gives the assurance that these rules are void of content insofar as they imply no factual propositions. On the other hand, also, vice-versa, it will turn out below that the feasibility of the nominalistic program implies the feasibility of the formalistic program. (For very lucid expositions of the philosophical aspects of this nominalistic view, see Hahn 1935 or Carnap 1935a, 1955b.) It might be doubted whether this (nominalistic) view should at all be subsumed under the view that considers mathematics to be a free creation of the mind, because it denies altogether the existence of mathematical objects. Moreover, the relationship between the two is extremely close, since also under the other view the so-called existence of mathematical objects consists solely in their being constructed in thought, and nominalists would not deny that we actually imagine (non-existent) objects behind the mathematical symbols and that these subjective ideas might even furnish the guiding principle in the choice of the syntactical rules.

The double vertical lines indicate material marked in the manuscript “Omit fromhere to p. 29”. Since this material was not crossed out, a plausible conjecture is that itwas to be omitted only from his oral presentation, But other conjectures are possiblefor example, that he came at a later time to think it duplicative of or superseded bydiscussions in *1953/9. See also the textual notes.

Gödel writes “demonstrat.-def.” in some other places without the hyphen. It might almost equally plausibly be read as “demonstrative definition” (which would be stylistically more attractive).

What indication there is as to what he has in mind is given by the following passage:

Of course it is to be noted that a demonstrat[ion]-def[inition] does not mean pointing the finger to the object for which a name is introduced (which in most cases is not possible even for physical concepts), but that it rather means explaining the meaning of a word by means of the situations in which it is used.

(From Gödel’s footnote 58. This note is flagged in the alternate version of the text printed from p. 29 [see the textual notes] and also in Gödel’s footnote 26, to which we have found no reference in the text.)

*The double vertical lines indicate material marked in the manuscript “Omit from here to p. 29”. Since this material was not crossed out, a plausible conjecture is that it was to be omitted only from his oral presentation. But other conjectures are possible, for example, that he came at a later time to think it duplicative or superseded by discussions in 1959/9. See also the textual notes.

As to the requirement of consistency, see fn. [23?]

It is also possible that Gödel intended to write a new note on this subject. In the manuscript, the text as we give it is above some crossed-out text in which something is said about the "requirement of consistency", which, however, he may have thought repeated points in (our) note 23.

See Ramsey 1926, pp. 368 and 382, and Carnap 1937, pp. 39 and 110. It is worth mentioning that Ramsey even succeeds in reducing them to explicit tautologies ( a = a ) by means of explicit definitions (see p. [24?] above), but at the expense of admitting propositions of infinite (and even transfinite) length, which of course entails the necessity of presupposing transfinite set theory in order to be able [to] deal with these infinite entities. Carnap confines himself to propositions of finite length, but instead has to consider infinite sets, sets of sets, etc., of these finite propositions.

For example, any axiom system ( S ) for set theory belonging to the series explained in the beginning of this lecture, the axiom of choice included, can be proved consistent by means of the axiom of the next order (or by means of the axiom that ( S ) is consistent) without the axiom of choice. Similarly, it is not impossible that the axioms of the lower levels of this hierarchy could be proved consistent by means of axioms of higher levels, with such restrictions, however, as would make them acceptable to intuitionists.

Examples of such abstract concepts are, for example, “set”, “function of integers”, “demonstrable” (the latter in the non-formalistic sense of “knowable to be true”), “derivable”, etc., or finally “there is”, referring to all possible combinations of symbols. The necessity of such concepts for the consistency proof of classical mathematics results from the fact that symbols can be mapped onto the integers, and therefore finitistic (and a fortiori, classical) number theory contains all proofs based solely upon them. The evidence for this fact so far is not absolutely conclusive because the evident axioms referring to the non-abstract concept under consideration have not been investigated thoroughly enough. However, the fact itself is acknowledged even by leading formalists; see [Bernays 1941a, pp. 144, 147; 1935, pp. 68, 69; 1935b, p. 94; 1954, p. 2; also Centzen 1937, p. 203].

The objection raised here against a syntactical foundation of number theory is substantially the same [as the one] which Poincaré leveled against both Frege’s and Hilbert’s foundation of number theory. However, this objection is not justified against Frege, because the logical concepts and axioms he has to presuppose do not explicitly contain the concept of a “finite manifold” with its axioms, while the grammatical concepts and considerations necessary to set up the syntactical rules and establish their tautological character do.

An unnumbered remark cited at this point appears at the bottom of page 29^1 of Gödel’s manuscript text. Neither a true footnote nor a textual insertion, it is rather a shorthand annotation. For a transcription and translation of its contents, see the textual notes.

This holds good also for those parts of mathematics which can be reduced to syntactic rules (see above). For these rules are based on the idea of a finite manifold (namely, of a finite sequence of symbols), and this idea and its properties are entirely independent of our free choice. In fact, its theory is equivalent to the theory of [the] integers. The possibility of so constructing a language that this theory is incorporated into it in the form of syntactic rules proves nothing. See fn. [?]h.

A conjecture as to what Gödel is referring to is that it is his footnote 35, to which there is no reference in the text. It reads as follows:

To be more exact the true situation as opposed to the view criticized is the following:

The meanings of mathematical terms are not reducible to the linguistic rules about their use except for a very restricted domain of mathematics (cf. [pp. 25–27?]).
Even where such a reduction is possible the linguistic rules cannot be considered to be something man-made and propositions about them to be lacking objective content because these rules are based on the idea of a finite manifold (in the form of finite sequences of symbols) and this idea (with all its properties) is entirely independent of any convention and free choice (hence is something objective). In fact, its theory is equivalent to arithmetic.

It could be, however, that this note was superseded by our note 29 (Gödel’s 49).

The rest of this page and part of the next are crossed out in Gödel’s manuscript.

See Darboux 1912[, p. 142]. The passage quoted continues as follows:

qui ne semblent distincts qu’à cause de la faiblesse de notre esprit, qui ne sont pour une pensée plus puissante qu’une seule et même chose, et dont la synthèse se révèle partiellement dans cette merveilleuse correspondance entre les Mathématiques abstraites d’une part, l’Astronomie et toutes les branches de la Physique de l’autre.
[and appear different only because of the weakness of our mind; but, for a more powerful intelligence, they are one and the same thing, whose synthesis is partially revealed in that marvelous correspondence between abstract mathematics on the one hand and astronomy and all branches of physics on the other.]

So here Hermite seems to turn toward Aristotelian realism. However, he does so only figuratively, since Platonism remains the only conception understandable for the human mind.