Proofs of the Cantor-Bernstein Theorem a Mathematical Excursion by Arie Hinkis

Format:

Book

Author/Creator:

Hinkis, Arie

Series:

Science networks historical studies 45

Science Networks. Historical Studies 45

Language:

English

Subjects (All):

Set theory.

Differential equations.

Logic, Symbolic and mathematical.

Mathematics.

Algebra.

algebra.

Physical Description:

1 online resource

Place of Publication:

Basel Springer Basel Imprint Birkhäuser 2013

Language Note:

English

Summary:

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

Contents:

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

Notes:

Includes bibliographical references and indexes

Other Format:

Printed edition:

ISBN:

9783034802246

3034802242

1299337465

9781299337466

OCLC:

841800168

Access Restriction:

Restricted for use by site license