vol 4: Glossary
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Cardinal number
[Hazewinkel, Cardinal number] [Borowski, Cardinal number]
Cantor defines cardinal number as follows:
Every [set] M has a definite #title "natural theology > VI glossary > power" #metaDescription "natural, religion, glossary, power" #metaKeywords "natural, religion, theology, glossary, power,motion, zeno, parmenides, aristotle, spatial, relationship, calculus, " #volHead "vol 4: Glossary" #capHead "" #pageHead "" #topPath "../" #nextURL "" #nextText "" #previousURL "aGlossaryToc.html" #previousText "Glossary: Toc"
power
[Hazewinkel, power] [Borowski, power] which we will also call its "cardinal number".
We will call by the name #title "natural theology > VI glossary > power" #metaDescription "natural, religion, glossary, power" #metaKeywords "natural, religion, theology, glossary, power,motion, zeno, parmenides, aristotle, spatial, relationship, calculus, " #volHead "vol 4: Glossary" #capHead "" #pageHead "" #topPath "../" #nextURL "" #nextText "" #previousURL "aGlossaryToc.html" #previousText "Glossary: Toc"
power
[Hazewinkel, power] [Borowski, power] or "cardinal number" of M the general concept which, by means of our active faculty of thought, arises when we make abstraction of the nature of the various elements of m and the order in which they are given. ... Cantor page 86.
The cardinal number of a set, therefore, is the number of units in it, regardless of their nature. This is, in a way, as far abstraction can go, and so gives us a very broad perspective on the world. We observe that the world has distinct parts. Practically, something is distinct if we can move it independently of other things.
Two sets are said to have the same cardinal number (or power, or to be equivalent to one another) if there elements can be placed in one-to-one correspondence.
The cardinal number of any finite set is the largest member of the ordered set natural numbers N that corresponds to it. Ordering makes it easier to establish a correspondence between two sets and to be certain that it is one-to-one. When counting sheep, for instance, we generally run them through a gate or race so that they are spread out in an ordered line with each sheep distinct from the others.
Cardinal number is a wider concept than natural number. The cardinal number of the set of natural numbers, card N, is aleph(0), the first transfinite cardinal. The existence of the larger transfinite cardinals, aleph(1), aleph(2) etc is established by Cantor's theorem.
Further reading
Books
| Borowski, Ephraim J, and Jonathan M Borwein, HarperCollins Dictionary of Mathematics, Harper Collins 1991 '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 |
| Burton, David M, Elementary Number Theory, Allyn and Bacon 1976 Preface: '"Plato said God is a geometer. Jacobi changed this to God is an arithmetician. Then came Kronecker and fashioned the memorable expression, God created the natural numbers, and all the rest is the work of man. Felix Klein." The purpose of the present volume is to give a simple account of classical number theory, as well as to impart some of the historical background in which the subject evolved.' 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 |
| Frege, Gottlob, The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Nature of Number, Northwestern UP 1980 Jacket: 'The book represents the first philosophically sound discussion of the concept of number in Western civilisation. It influenced profoundly developments in the philosophy of mathematics, general ontology and mathematics.' Amazon back |
| Hazewinkel, Michiel, and (managing editor), 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 |
| Jech, Thomas, Set Theory, Springer 1997 Jacket: 'This book covers major areas of modern set theory: cardinal arithmetic, constructible sets, forcing and Boolean-valued models, large cardinals and descriptive set theory. ... It can be used as a textbook for a graduate course in set theory and can serve as a reference book.' Amazon back |
| Saharon, Shelah, Cardinal Arithmetic, Clarendon Press 1994 Introduction: 'Cantor should have no problems understanding and (so I feel) appreciating the theorems and most of the proofs in this book. In particular the book consists of absolute theorems and assumes no more knowledge than of basic set theory. This is not done, however, on the expense of being isolated, or irrelevant, to the mainstream of the field. Cardinal arithmetic has been the central problem in the development of set theory.' page xi Amazon back |
Links
| Bert G Wachsmuth A hierarchy of infinity - cardinal numbers 'Definition of Cardinal Number Two sets A and B are called equivalent if there exists a bijection between A and B. The two sets are said to have the same cardinality, or power. The cardinality of a set A is denoted by card(A). The cardinal numbers of two sets are equal if the sets are equivalent.' back |
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