Proofs of the Cantor-Bernstein Theorem

This 'excursion' through the evolution of research mathematics takes in some 40 papers, published up to the 1970s, on proofs of the Cantor-Bernstein theorem. It provides a detailed example of proof-processing in the derivation of Bernstein's division theorem.

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.

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" Includes supplementary material: sn.pub/extras

Inhalt

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. Knig'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.- Sierpiski'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.