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Library | Item Barcode | Call Number | Material Type | Item Category 1 | Status |
|---|---|---|---|---|---|
Searching... | 30000003196890 | QA9 V54 1995 | Open Access Book | Book | Searching... |
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Summary
Summary
The concept of infinity is one of the most important, and at the same time, one of the most mysterious concepts of science. Already in antiquity many philosophers and mathematicians pondered over its contradictory nature. In mathematics, the contradictions connected with infinity intensified after the creation, at the end of the 19th century, of the theory of infinite sets and the subsequent discovery, soon after, of paradoxes in this theory. At the time, many scientists ignored the paradoxes and used set theory extensively in their work, while others subjected set-theoretic methods in mathematics to harsh criticism. The debate intensified when a group of French mathematicians, who wrote under the pseudonym of Nicolas Bourbaki, tried to erect the whole edifice of mathematics on the single notion of a set. Some mathematicians greeted this attempt enthusiastically while others regarded it as an unnecessary formalization, an attempt to tear mathematics away from life-giving practical applications that sustain it. These differences notwithstanding, Bourbaki has had a significant influence on the evolution of mathematics in the twentieth century. In this book we try to tell the reader how the idea of the infinite arose and developed in physics and in mathematics, how the theory of infinite sets was constructed, what paradoxes it has led to, what significant efforts have been made to eliminate the resulting contradictions, and what routes scientists are trying to find that would provide a way out of the many difficulties.
Reviews 1
Choice Review
The four distinct chapters in this book each require a bit more mathematical sophistication. The beginning addresses the early acceptance of infinity as a function of the beliefs of the great minds of a given era, and ends with the development of the notion of curved space and its ramifications. Leaving reality behind, a review of standard set theoretic results and problems are addressed to prepare the reader for Cantor's diagonalization process, which produces the irrefutable result that there must be at least two distinct infinities. Some of the repercussions of this, such as the existence of transcendental numbers, are examined. Remaining chapters address many of the mathematical consequences resulting from the existence of countable sets and the continuum. Vilenkin confronts nowhere-continuous functions, uncountable sets of measure zero, space filling curves, and the realization that the distinction between curve, surface, and solid is not nearly so clear. Well suited for any college library; easy to read with intuitive explanations of some difficult concepts. Both students and instructors will benefit. General; undergraduate. J. R. Burke; Gonzaga University
