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.

zlib/no-category/Hinkis, Arie/Proofs of the Cantor-Bernstein Theorem ||_53210449.pdf

by Arie Hinkis

Hinkis, Arie

Springer Nature Switzerland AG

Birkhäuser/Springer

Science Networks. Historical Studies -- 45, Basel, Switzerland, 2013

Science networks historical studies, v. 45, Heidelberg, 2013

Springer Nature, New York, 2013

Switzerland, Switzerland

sm20815463

MiU

Preface. - Part I: Cantor and Dedekind
Cantor's CBT proof for sets of the power of (II)
Generalizing Cantor's CBT proof
CBT in Cantor's 1878 Beitrag
The theory of inconsistent sets
Comparability in Cantor's writings
The scheme of complete disjunction
Ruptures in the Cantor-Dedekind correspondence
The inconsistency of Dedekind's infinite set
Dedekind's proof of CBT
Part II: The early proofs
Schröder's Proof of CBT
Bernstein, Borel and CBT
Schoenflies' 1900 proof of CBT
Zermelo's 1901 proof of CBT
Bernstein's Division Theorem
Part III: Under the logicist sky
Russell's 1902 proof of CBT
The role of CBT in Russell’s Paradox
Jourdain's 1904 generalization of Grundlagen
Harward 1905 on Jourdain 1904
Poincaré and CBT
Peano's proof of CBT
J. Kőnig's strings gestalt
From kings to graphs
Jourdain's improvements round
Zermelo's 1908 proof of CBT
Korselt's proof of CB
Proofs of CBT in Principia Mathematica
The origin of Hausdorff Paradox in BDT
Part IV: At the Polish school
Sierpiński's proofs of BDT
Banach's proof of CBT
Kuratowski's proof of BDT
Early fixed-point CBT proofs: Whittaker; Tarski-Knaster
CBT and BDT for order-types
Sikorski's proof of CBT for Boolean algebras
Tarski's proofs of BDT and the inequality-BDT
Tarski's Fixed-Point Theorem and CBT
Reichbach's proof of CBT
Part V: Other ends and beginnings
Hellmann's proof of CBT
CBT and intuitionism
CBT in category theory
Conclusion
Bibliography
Index of names
Index of subjects.

2013-03-12