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Proofs of the Cantor-Bernstein Theorem


Proofs of the Cantor-Bernstein Theorem

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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