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NATURAL FORMALIZATION: DERIVING THE CANTOR-BERNSTEIN THEOREM IN ZF

Part of: General logic Artificial intelligence (68Txx) Proof theory and constructive mathematics

Published online by Cambridge University Press:  18 November 2019

WILFRIED SIEG  and
PATRICK WALSH
WILFRIED SIEG
Affiliation:
DEPARTMENT OF PHILOSOPHY CARNEGIE MELLON UNIVERSITYPITTSBURGH, PA15213, USAE-mail: sieg@cmu.edu
PATRICK WALSH
Affiliation:
DEPARTMENT OF PHILOSOPHY CARNEGIE MELLON UNIVERSITY QATARDOHA, QATARE-mail: pwalsh@andrew.cmu.edu
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Abstract

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

Keywords

verification in set theorynatural deductionproof analysisreasoning with gapshuman-oriented theorem proving

MSC classification

Primary: 03F07: Structure of proofs
Secondary: 03B35: Mechanization of proofs and logical operations 68T15: Theorem proving (deduction, resolution, etc.)
Type
Research Article
Information
The Review of Symbolic Logic , Volume 14 , Issue 1 , March 2021 , pp. 250 - 284
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Copyright
© Association for Symbolic Logic, 2019

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