## Proofs of the Cantor-Bernstein Theorem

June 9, 2013 | 2:29 am | 0 Comment | 950 Views**Proofs of the Cantor-Bernstein Theorem
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**Book Details**

Hardcover: 452 pages

Publisher: Birkhäuser; 2013 edition (February 22, 2013)

Language: English

ISBN-10: 3034802234

ISBN-13: 978-3034802239

File Size: 2.9 Mb | File Format: PDF

**Book Description**

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**

Chapter 1: Cantor´s CBT Proof for Sets of the Power of (II)

Chapter 2: Generalizing Cantor´s CBT Proof

Chapter 3: CBT in Cantor´s 1878 Beitrag

Chapter 4: The Theory of Inconsistent Sets

Chapter 5: Comparability in Cantor´s Writings

Chapter 6: The Scheme of Complete Disjunction

Chapter 7: Ruptures in the Cantor-Dedekind Correspondence

Chapter 8: The Inconsistency of Dedekind´s Infinite Set

Chapter 9: Dedekind´s Proof of CBT

Chapter 10: Schröder´s Proof of CBT

Chapter 11: Bernstein, Borel and CBT

Chapter 12: Schoenflies´ 1900 Proof of CBT

Chapter 13: Zermelo´s 1901 Proof of CBT

Chapter 14: Bernstein´s Division Theorem

Chapter 15: Russell´s 1902 Proof of CBT

Chapter 16: The Role of CBT in Russell´s Paradox

Chapter 17: Jourdain´s 1904 Generalization of Grundlagen

Chapter 18: Harward 1905 on Jourdain 1904

Chapter 19: Poincaré and CBT

Chapter 20: Peano´s Proof of CBT

Chapter 21: J. Konig´s Strings Gestalt

Chapter 22: From Kings to Graphs

Chapter 23: Jourdain´s Improvements Round

Chapter 24: Zermelo´s 1908 Proof of CBT

Chapter 25: Korselt´s Proof of CBT

Chapter 26: Proofs of CBT in Principia Mathematica

Chapter 27: The Origin of Hausdorff Paradox in BDT

Chapter 28: Sierpinski’s Proofs of BDT

Chapter 29: Banach´s Proof of CBT

Chapter 30: Kuratowski´s Proof of BDT

Chapter 31: Early Fixed-Point CBT Proofs: Whittaker; Tarski-Knaster

Chapter 32: CBT and BDT for Order-Types

Chapter 33: Sikorski´s Proof of CBT for Boolean Algebras

Chapter 34: Tarski´s Proofs of BDT and the Inequality-BDT

Chapter 35: Tarski´s Fixed-Point Theorem and CBT

Chapter 36: Reichbach´s Proof of CBT

Chapter 37: Hellmann´s Proof of CBT

Chapter 38: CBT and Intuitionism

Chapter 39: CBT in Category Theory

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