The chief purpose of the book is to present, in detail, a
compilation of proofs of the Cantor-Bernstein Theorem (CBT)
published through the years since the 1870's. Over thirty such
proofs are surveyed.
The book comprises five parts. In the first part the discussion
covers the role of CBT and related notions in the writings of
Cantor and Dedekind. New views are presented, especially regarding
the general proof of CBT obtained by Cantor, his proof of the
Comparability Theorem, the ruptures in the Cantor-Dedekind
correspondence and the origin of Dedekind's proof of CBT.
The second part covers the first CBT proofs published
(1896-1901). The works of the following mathematicians is
considered in detail: Schroder, Bernstein, Bore, Schoenflies and
Zermelo. Here a subtheme of the book is launched; it concerns the
research project following Bernstein's Division Theorem (BDT).
In its third part the book covers proofs that emerged during the
period when the logicist movement was developed (1902-1912). It
covers the works of Russell and Whitehead, Jourdain, Harward,
Poincare, J. Konig, D. Konig (his results in graph theory), Peano,
Zermelo, Korselt. Also Hausdorff's paradox is discussed linking it
to BDT.
In the fourth part of the book are discussed the developments of
CBT and BDT (including the inequality-BDT) in the hands of the
mathematicians of the Polish School of Logic, including Sierpi ski,
Banach, Tarski, Lindenbaum, Kuratowski, Sikorski, Knaster, the
British Whittaker, and Reichbach.
Finally, in the fifth part, the main discussion concentrates on
the attempts to port CBT to intuitionist mathematics (with results
by Brouwer, Myhill, van Dalen and Troelstra) and to Category Theory
(by Trnkova and Koubek).The second purpose of the book is to
develop a methodology for the comparison of proofs. The core idea
of this methodology is that a proof can be described by two
descriptors, called gestalt and metaphor. It is by comparison of
their descriptors that the comparison of proofs is obtained. The
process by which proof descriptors are extracted from a proof is
named 'proof-processing', and it is conjectured that mathematicians
perform proof-processing habitually, in the study of proofs.
General
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