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Birkhäuser - Birkhäuser History of Science | Proofs of the Cantor-Bernstein Theorem - A Mathematical Excursion

Proofs of the Cantor-Bernstein Theorem

A Mathematical Excursion

Hinkis, Arie

2013, XXIII, 429 p. 29 illus., 3 illus. in color.

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  • A scenic-view excursion through the developmental area of research mathematics around the Cantor-Bernstein Theorem (CBT) and related results stemming from Bernstein's Division Theorem (BDT), with emphasis on providing accurate proofs, similar to the originals, enabling appreciation of the historical context
  • Suggests a revision in the common understanding of central points in Cantor's set theory, such as his proofs of CBT and of the Comparability Theorem, the nature of the Limitation Principle, inconsistent sets and set theory by 1877
  • Opens up a discussion on the methodology of proof comparison, through the proof descriptors 'gestalt' and 'metaphor', generated by proof-processing
  • Resembles, in its attempt to present a diachronic narrative of one mathematical topic, from a methodological perspective, the celebrated book of Lakatos "Proofs and Refutations"

This book offers an excursion through the developmental area of research mathematics. It presents some 40 papers, published between the 1870s and the 1970s, on proofs of the Cantor-Bernstein theorem and the related Bernstein division theorem. While the emphasis is placed on providing accurate proofs, similar to the originals, the discussion is broadened to include aspects that pertain to the methodology of the development of mathematics and to the philosophy of mathematics. Works of prominent mathematicians and logicians are reviewed, including Cantor, Dedekind, Schröder, Bernstein, Borel, Zermelo, Poincaré, Russell, Peano, the Königs, Hausdorff, Sierpinski, Tarski, Banach, Brouwer and several others mainly of the Polish and the Dutch schools. In its attempt to present a diachronic narrative of one mathematical topic, the book resembles Lakatos’ celebrated book Proofs and Refutations. Indeed, some of the observations made by Lakatos are corroborated herein. The analogy between the two books is clearly anything but superficial, as the present book also offers new theoretical insights into the methodology of the development of mathematics (proof-processing), with implications for the historiography of mathematics.

Content Level » Research

Keywords » Cantor-Bernstein theorem - gestalt in mathematics - history of science - metaphor in mathematics - methodology of mathematics - philosophy of science - proof-processing

Related subjects » Birkhäuser History of Science - Birkhäuser Mathematics

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