The Strength of Nonstandard Analysis
Nonstandard Analysis enhances mathematical reasoning by introducing new ways of expression and deduction. Distinguishing between standard and nonstandard mathematical objects, its inventor, the eminent mathematician Abraham Robinson, settled in 1961 the c
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SpringerWienNewYork
Imme van den Berg Vitor Neves (eds.) The Strength of Nonstandard Analysis
SpringerWienNewYork
Imme van den Berg Departamento de Matematica Universidade de Evora, Evora, Portugal
Vitor Neves Departamento de Matematica Universidade de Aveiro, Aveiro, Portugal
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With 16 Figures Library of Congress Control Number 2006938803
ISBN-10 3-211-49904-0 SpringerWienNewYork ISBN-13 978-3-211-49904-7 SpringerWienNewYork
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Forty-five years ago, an article appeared in the Proceedings of the Royal Academy of Sciences of the Netherlands Series A, 64, 432-440 and Indagationes Math. 23 (4), 1961, with the mysterious title "Non-standard Analysis" authored by the eminent mathematician and logician Abraham Robinson (1908-1974). The title of the paper turned out to be a contraction of the two terms "Nonstandard Model" used in model theory and "Analysis". It presents a treatment of classical analysis based on a theory of infinitesimals in the context of a non-standard model of the real number system R. In the Introduction of the article, Robinson states: "It is our main purpose to show that the models provide a natural approach to the age old problem of producing a calculus involving infinitesimal (infinitely small) and infinitely large quantities. As is well-known the use of infinitesimals strongly advocated by Leibniz and unhesitatingly accepted by Euler fell into disrepute after the advent of Cauchy's methods which put Mathematical Analysis on a firm foundation". To bring out more clearly the importance of Robinson's creation of a rigorous theory of infinitesimals and their reciprocals, the infinitely
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