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 24.
Pàgina 4
... sequences were studied , and probability theory had events , but one did not operate on these things ( at least not beyond elementary operations given by ' and ' , ' or ' , ' not ' ) , so there was no need for a general notion of set ...
... sequences were studied , and probability theory had events , but one did not operate on these things ( at least not beyond elementary operations given by ' and ' , ' or ' , ' not ' ) , so there was no need for a general notion of set ...
Pàgina 5
... sequence exist , then you conclude that there is also an infinitely small member ; in my opinion it only implies that actually every arbitrary , finite , definable fraction of every arbitrarily small size exists . ” Bernoulli replied to ...
... sequence exist , then you conclude that there is also an infinitely small member ; in my opinion it only implies that actually every arbitrary , finite , definable fraction of every arbitrarily small size exists . ” Bernoulli replied to ...
Pàgina 6
... sequence , we can surmise that Bolzano wanted to found the 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 ...
... sequence , we can surmise that Bolzano wanted to found the 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 ...
Pàgina 10
... sequence of intervals ( a , b ) . There are two possibilities : n ( i ) lim a1 = lim b1 = r , then r & S because otherwise ra1 or r = b1 for some k , which is impossible as až < r < by for all k . k ( ii ) a = lim a , < lim b1 = b . In ...
... sequence of intervals ( a , b ) . There are two possibilities : n ( i ) lim a1 = lim b1 = r , then r & S because otherwise ra1 or r = b1 for some k , which is impossible as až < r < by for all k . k ( ii ) a = lim a , < lim b1 = b . In ...
Pàgina 15
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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