vol 4: Glossary
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Cantor's theorem
[Hazewinkel, Cantor theorem] [Borowski, Cantor's diagonal theorem]
Cantor proved, using the famous diagonal process that the real numbers between 0 and 1 could not be put into one-one correspondence with the natural numbers, ie that they are uncountable. But is this the end of the story?
The central specific problem which occupied Cantor was the celebrated continuum problem posed by him in 1878, i.e. the problem of whether there are more than two infinite sizes represented in the real line. Hallett, x.
In 1895 Cantor wrote
After we have introduced the least transfinite cardinal number aleph(0) [the cardinal number of the set of natural numbers] and derived its properties that lie the most readily to hand, the question arises as to the higher cardinal numbers and how they proceed from aleph(0). We shall show that the transfinite cardinal numbers can be arranged according to their magnitude, and, in this order, form, like the finite numbers, a "well-ordered aggregate" in an extended sense of the words. Out of aleph(0) proceeds, by a definite law, the next greater cardinal number aleph(1), out of this by the same law the next greater aleph(2), and so on. But even the unlimited sequence of cardinal numbers
aleph(0), aleph(1), aleph(2), ... , aleph(nu), ... does not exhaust the conception of transfinite cardinal number. We will prove the existence of a cardinal number, which we will denote by aleph(omega) and which shows itself to be the next greater to all the numbers aleph(nu); out of it proceeds in the same way as aleph(1) out of aleph(0) a next greater aleph(omega+1), and so on without end. Cantor, 108.
A modern statement and proof of this theorem depends assumes the existence of the power set of a set S. The power set P(S) of S is the set of all subsets of S.
Cantor's Theorem: To any set S, there exists a set of greater cardinal number, in particular, card P(S) > card S.
Proof: On the one hand, it is clear that there is an injective mapping of S into P(S), namely the one mapping the elements a of S to the singleton sets {a} of P(S).
Now we show that there is no map F from S to P(S) which covers P(S), i.e. there are elements of P(S) that are not images of elements of S.
We define a set U = {x is an element of S and x is not an element of its image under F, F(x)}, and show that U is never an image under F.
Suppose to the contrary, that U = F(u) for some u in S. Now either u is an element of U, or it is not.
Now if u is an element of U, then u is an element of F(u), since we have supposed that U = F(u). But U, by definition, contains only those elements of S that are not elements of their images under F. So the assumption that u is an element of U leads to a contradiction.
On the other hand, if u is not an element of U, u is not an element of F(u) either. But, since U by definition contains all the elements of S that are not elements of their images F(u), this implies that u is an element of U, another contradiction.
These contradictions show that the supposition that U = F(u) is untenable, There are elements in P(S) that correspond to no element in S, hence card P(S) > card S. Gellert, 327.
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 |
| Gellert, Walter, et al (eds), The VNR Concise Encyclopedia of Mathematics , Van Nostrand Reinhold, 1994. Preface: '... there is a wide demand for a survey of the results of mathematics ... Our task was to describe mathematical interrelations as briefly and precisely as possible. ... Colours are used extensively to help the reader. ... Ample examples help to make general statements understandable. ... A systematic subdivision of the material, many brief section headings, and tables are meant to provide the reader with quick and reliable orientation. The detailed index to the book gives easy access to specific questions. ...' The Editors and Publishers 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 |
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