Sets and integration An outline of the developmentSpringer Science & Business Media, 6 de des. 2012 - 162 pàgines The present text resulted from lectures given by the authors at the Rijks Universiteit at Utrecht. These lectures were part of a series on 'History of Contemporary Mathematics'. The need for such an enterprise was generally felt, since the curriculum at many universities is designed to suit an efficient treatment of advanced subjects rather than to reflect the development of notions and techniques. As it is very likely that this trend will continue, we decided to offer lectures of a less technical nature to provide students and interested listeners with a survey of the history of topics in our present-day mathematics. We consider it very useful for a mathematician to have an acquaintance with the history of the development of his subject, especially in the nineteenth century where the germs of many of modern disciplines can be found. Our attention has therefore been mainly directed to relatively young developments. In the lectures we tried to stay clear of both oversimplification and extreme technicality. The result is a text, that should not cause difficulties to a reader with a working knowledge of mathematics. The developments sketched in this book are fundamental for many areas in mathematics and the notions considered are crucial almost everywhere. The book may be most useful, in particular, for those teaching mathematics. |
Des de l'interior del llibre
Resultats 1 - 5 de 27.
Pàgina 6
... concept of sequence on the successor - relation only . Once the concept of sequence is available Bolzano introduces countable and finite sets ( zähl- bare Vielheiten ) and integers ( ganze Zahlen ) as sequences with a first element and ...
... concept of sequence on the successor - relation only . Once the concept of sequence is available Bolzano introduces countable and finite sets ( zähl- bare Vielheiten ) and integers ( ganze Zahlen ) as sequences with a first element and ...
Pàgina 8
... concepts under which nothing falls and exploits these for the definition of the number 0 [ 4 ] . Frege , in his " Grundgesetzen der Arithmetik " [ 42 ] , introduction , rightly critisizes Dedekind for not consistently identifying ...
... concepts under which nothing falls and exploits these for the definition of the number 0 [ 4 ] . Frege , in his " Grundgesetzen der Arithmetik " [ 42 ] , introduction , rightly critisizes Dedekind for not consistently identifying ...
Pàgina 9
... concepts of set theory that now belong to the standard part of any elementary text- book of mathematics . In particular the signs E , ~ , U are used in their present - day meaning . The early history of CANTOR ( cf. [ 25 ] , [ 4 ] ...
... concepts of set theory that now belong to the standard part of any elementary text- book of mathematics . In particular the signs E , ~ , U are used in their present - day meaning . The early history of CANTOR ( cf. [ 25 ] , [ 4 ] ...
Pàgina 11
... concepts , we also meet the Cantor - Bernstein theorem ( [ 25 ] , p . 201 ) . Cantor states the theorem for sets of cardinality of the second number class and concludes it as a corrollary of the following theorem : a subset of the ...
... concepts , we also meet the Cantor - Bernstein theorem ( [ 25 ] , p . 201 ) . Cantor states the theorem for sets of cardinality of the second number class and concludes it as a corrollary of the following theorem : a subset of the ...
Pàgina 14
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Continguts
7 | |
The paradoxes | 21 |
Zermelo takes over | 34 |
Making inconsistent sets respectable | 45 |
The consistency of the axiom of choice and the continuum hypothesis | 52 |
Large cardinals | 63 |
Introduction the period before Riemann | 77 |
Riemann Lebesgue real functions | 91 |
Modern theory of the integral | 133 |
Bibliography | 149 |
Index | 155 |
List of Mathematicians | 161 |
Altres edicions - Mostra-ho tot
Sets and Integration An Outline of the Development D. van Dalen,A. F. Monna Visualització de fragments - 1972 |
Frases i termes més freqüents
algebra analysis axiom of choice axiom of foundation axiomatic set theory Baire Bolzano Borel bounded variation Burali-Forti calculable called Cantor Carathéodory cardinals Cohen concept considered constructive continuous functions continuum hypothesis countable Dedekind defined definition Denjoy denoted dérivée differential domain element ensemble equivalent example existence finite formulated Fraenkel function f geometry Georg Cantor Gödel Grundlagen Heyenoort 61 Hilbert infinity instance interval introduced König Lebesgue Lebesgue measurable Lebesgue-integral Leçons linear logic mapping Math mathematicians mathematics mathématique Measurable cardinals measure theory Mengenlehre mention method modern natural numbers Neumann nombre notation notion objects one-one ordered sets ordinals paper paradox Peano Poincaré polyhedrons principle problem proof properties proved real functions real numbers remarks Riemann Riemann-integral second number class sequence set of reals Skolem space subset Tarski theorem topology transfinite urelements valeurs variable VOLTERRA well-ordered Zermelo ZF is consistent