Science Networks. Historical Studies

Proofs of the Cantor-Bernstein Theorem

A Mathematical Excursion

Authors: Hinkis, Arie

  • 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"
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  • ISBN 978-3-0348-0224-6
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About this book

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.

Table of contents (39 chapters)

Buy this book

eBook $99.00
price for USA (gross)
  • ISBN 978-3-0348-0224-6
  • Digitally watermarked, DRM-free
  • Included format: EPUB, PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $129.00
price for USA
  • ISBN 978-3-0348-0223-9
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Rent the ebook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
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Bibliographic Information

Bibliographic Information
Book Title
Proofs of the Cantor-Bernstein Theorem
Book Subtitle
A Mathematical Excursion
Authors
Series Title
Science Networks. Historical Studies
Series Volume
45
Copyright
2013
Publisher
Birkhäuser Basel
Copyright Holder
Springer Basel
eBook ISBN
978-3-0348-0224-6
DOI
10.1007/978-3-0348-0224-6
Hardcover ISBN
978-3-0348-0223-9
Series ISSN
1421-6329
Edition Number
1
Number of Pages
XXIII, 429
Number of Illustrations and Tables
21 b/w illustrations, 3 illustrations in colour
Topics