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Analysis 0. Ein logischer Einstieg in die Analysis

Analysis 0. Ein logischer Einstieg in die Analysis

          
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About the Book

Was bedeutet eigentlich das Gleichheitszeichen? Was ist ein Beweis? Warum ist ungerade mal ungerade ungerade, aber minus mal minus plus? Auf solche Grundlagen gehen die Lehrveranstaltungen zur Analysis an unseren Hochschulen selten ein. In Wolfgang Helbigs Einführung finden sich Antworten auf Fragen wie diese.Anders als in vielen Einführungen geht Helbig nicht nur auf die reellen und komplexen, sondern auch auf die natürlichen, ganzen und rationalen Zahlen ein. Sie werden durch Axiome so definiert, daß sie lückenlose Beweise nicht nur in der Analysis, sondern auch in der Zahlentheorie oder der Kombinatorik ermöglichen. Die Widerspruchsfreiheit dieser Axiome wird durch vollständig ausgeführte mengentheoretische Konstruktionen nachgewiesen, wobei einzig die Existenz einer Peano-Struktur vorausgesetzt wird.Die Eigenschaften stetiger Funktionen, konvergenter Folgen und Reihen, zusammenhängender und kompakter Mengen werden aus den Axiomen des metrischen Raums abgeleitet und als weiteres Beispiel der axiomatischen Methode auf Zahlenbereiche und Funktionenräume übertragen, um Potenzen mit reellen und komplexen Exponenten zu studieren.Alle Sätze, vom Prinzip der rekursiven Definition bis zum Satz von Heine-Borel, und alle Formeln, von der geometrischen Reihe bis zum Cauchyprodukt, werden allein aus den Axiomen abgeleitet. In Helbigs Analysis 0 gibt es nichts "Triviales", jeder Schluß wird begründet. Dabei legt Helbig besonderen Wert auf die Leserfreundlichkeit der Beweise.So kann der Leser die Chance wahrnehmen, die Schlüssigkeit der Argumentation selbst zu beurteilen. Auch der mathematisch Fortgeschrittene zieht aus der Lektüre Gewinn - findet doch auch er die Herleitung zahlreicher vertrauter Sätze, die er bislang beweislos und unkritisch benutzt hat, etwa zu Eigenschaften endlicher Mengen oder den Rechengesetzen endlicher Summen.


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Product Details
  • ISBN-13: 9783838203010
  • Publisher: Ibidem Press
  • Publisher Imprint: Ibidem Press
  • Height: 210 mm
  • No of Pages: 206
  • Spine Width: 11 mm
  • Width: 148 mm
  • ISBN-10: 3838203011
  • Publisher Date: 01 Sep 2011
  • Binding: Paperback
  • Language: German
  • Returnable: N
  • Weight: 249 gr


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