vol 4: Glossary
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Cantor's paradox
[Hazewinkel, Antinomy] [Borowski, Cantor's paradox]
An antinomy, contradiction or paradox is a situation in which two mutually contradictory statements (p and not-p) are demonstrated, each one having been deduced by means that are convincing from the point of view of the same theory.
Cantor's paradox was recognised by Cantor himself in 1899:
The cardinal number card Y of a set Y is defined to be the set of all sets X which are equinumerous with Y (ie for which there is a one-one correspondence between Y and X ). We define card Y =< card Z to mean that Y is equinumerous with a subset of Z; by card Y < card Z we mean card Y =< card Z and card Y is not equal to card Z.
Cantor proved that if P(Y) is the set of all subsets of Y, then card Y < card P(Y). (Cantor's Theorem)
Let C be the universal set, ie the set of all sets. Now P(C) is a subset of C, so it follows that card P(C) =< card C .
On the other hand, by Cantor's theorem, card C < card P(C).
The Schroeder-Bersten Theorem asserts that if card Y =< card Z and card Z =< card Y, then card Y = card Z. Hence card C = card P(C), contradicting Card C < card P(C). Mendelson, 2.
The existence of Cantor's paradox points to a weakness in the assumptions made in the derivation of the paradox. In particular, the assumption of an all inclusive (universal) set seems suspect. As Cantor himself proved, there appears to be no largest set.
Further reading
Books
| Borowski, Ephraim J, & Johnathan M Borwein, Collins Dictionary of Mathematics, Harper Collins 1089 'It is the immodest hope of the authors that this dictionary will not only prove valuable as a reference book for students of mathematics at all levels from secondary schools to a master's degree, but also offer much to interest a more general readership. Amazon back |
| Cantor, Georg, Contributions to the Founding of the Theory of Transfinite Numbers (Translated, with Introduction and Notes by Philip E B Jourdain), Dover 1955 Jacket: 'One of the greatest mathematical classics of all time, this work established a new field of mathematics which was to be of incalculable importance in topology, number theory, analysis, theory of functions, etc, as well as the entire field of modern logic.' Amazon back |
| Hallett, Michael, Cantorian set theory and limitation of size, Oxford UP 1984 Jacket: 'This book will be of use to a wide audience, from beginning students of set theory (who can gain from it a sense of how the subject reached its present form), to mathematical set theorists (who will find an expert guide to the early literature), and for anyone concerned with the philosophy of mathematics (who will be interested by the extensive and perceptive discussion of the set concept).' Daniel Isaacson. Amazon back |
| Hazewinkel, Michiel, Encyclopaedia of Mathematics (6 volumes), Kluwer Academic and Toppan 1995 'The Encyclopaedia of mathematics aims to be a reference work for all parts of mathematics. It is a translation with updates and editorial comments of the Soviet Mathematical Encyclopaedia published by 'Soviet Encyclopaedia Publishing House' in five volumes in 1977-85.' Amazon back |
| Hughes, Patrick, Vicious Circles and Infinity: A Panoply of Paradoxes, 1975 Amazon back |
| Mendelson, Elliott, Introduction to Mathematical Logic, van Nostrand 1987 Preface: '... a compact introduction to some of the principal topics of mathematical logic. ... In the belief that beginners should be exposed to the most natural and easiest proofs, free swinging set-theoretical methods have been used." http://www.amazon.com/exec/obidos/ASIN/0412069717/tnrp">Amazon back |
Links
| O'Connor and Robertson Cantor Hilbert described Cantor's work as:- ...the finest product of mathematical genius and one of the supreme achievements of purely intellectual human activity. back |
| Paul Golba Georg Cantor (1845-1918) back |
| PlanetMath.org Cantor's paradox 'Maths for the people by the people' back |
Click on an "Amazon" link in the booklist at the foot of the page to buy the book, see more details or search for similar items
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