Natural Formalization proposes a concrete way of expanding proof theory from the meta-mathematical investigation of formal theories to an examination of “the concept of the specifically mathematical proof.” Formal proofs play a role for this examination in as much as they reflect the essential structure and systematic construction of mathematical proofs. We emphasize three crucial features of our formal inference mechanism: (1) the underlying logical calculus is built for reasoning with gaps and for providing strategic directions, (2) the mathematical frame is a definitional extension of Zermelo–Fraenkel set theory and has a hierarchically organized structure of concepts and operations, and (3) the construction of formal proofs is deeply connected to the frame through rules for definitions and lemmas.To bring these general ideas to life, we examine, as a case study, proofs of the Cantor–Bernstein Theorem that do not appeal to the principle of choice. A thorough analysis of the multitude of “different” informal proofs seems to reduce them to exactly one. The natural formalization confirms that there is one proof, but that it comes in two variants due to Dedekind and Zermelo, respectively. In this way it enhances the conceptual understanding of the represented informal proofs. The formal, computational work is carried out with the proof search system AProS that serves as a proof assistant and implements the above inference mechanism; it can be fully inspected at http://www.phil.cmu.edu/legacy/Proof_Site/.We must—that is my conviction—take the concept of the specifically mathematical proof as an object of investigation.Hilbert 1918

中文翻译：

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自然形式化：在 ZF 中导出康托-伯恩斯坦定理

自然形式化提出了一种将证明理论从形式理论的元数学研究扩展到对“特定数学证明的概念”的检查的具体方法。形式证明在此考试中发挥作用，因为它们反映了数学证明的基本结构和系统构造。我们强调我们形式推理机制的三个关键特征：（1）底层逻辑演算是为有差距的推理并且对于提供战略方向，（2）数学框架是Zermelo-Fraenkel集合论的定义扩展，具有概念和操作的分层组织结构，（3）形式证明的构造通过定义和引理的规则与框架紧密相连。为了将这些一般想法变为现实，我们作为案例研究检查了不诉诸选择原则的康托尔-伯恩斯坦定理的证明。对大量“不同”的非正式证明进行彻底分析似乎可以将它们简化为一个。这自然形式化确认有一个证明，但由于 Dedekind 和 Zermelo 分别有两种变体。通过这种方式，它增强了对所表示的非正式证明的概念理解。正式的、计算的工作是使用证明搜索系统 AProS 进行的，它作为证明助手并实现了上述推理机制；它可以在http://www.phil.cmu.edu/legacy/Proof_Site/.我们必须——这是我的信念——将具体数学证明的概念作为研究对象。Hilbert 1918