NATURAL FORMALIZATION: DERIVING THE CANTOR-BERNSTEIN THEOREM IN ZF

Author:

SIEG WILFRIED,WALSH PATRICK

Abstract

AbstractNatural 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

Publisher

Cambridge University Press (CUP)

Subject

Logic,Philosophy,Mathematics (miscellaneous)

Cited by 4 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Proofs as Objects;Axiomatic Thinking I;2022

2. Human-Centered Automated Proof Search;Journal of Automated Reasoning;2021-07-30

3. GODEL'S PHILOSOPHICAL CHALLENGE (TO TURING);STUDIA SEMIOTYCZNE;2020

4. The Cantor–Bernstein theorem: how many proofs?;Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences;2019-01-21

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