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Infinity: Beyond the Beyond the Beyond

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  • Anonymous

    Posted September 4, 2008

    Infinity analyzed

    Nobody explains mathematical ideas for the layman as does Lillian R. Lieber. And the fanciful illustrations that always accompany her work, done by Hugh Gray Lieber, are amusing and informative. Infinity: Beyond the Beyond the Beyond presents an account of how mathematics has learned to deal with the infinite, through the work of Georg Cantor. Controversial in its day, Cantor's set theory and transfinite arithmetic are now part of the foundations of modern mathematics. Perhaps the most startling idea to be had from this book is that infinite sets are not all of the same size. I have before me a copy of the 1953 original, as well as the 2007 abridgement. Aside from the fact that the older book is a hardcover, the abridgement is the better book. The editor, Barry Mazur, a mathematician at Harvard, has removed the dated, nonmathematical introductory material and the chapters on calculus. This book is now a superb layman's guide to the mathematics of transfinities. If you would like more biography and less mathematics, you might try The Mystery of the Aleph, by Amir D. Aczel. Note: In 1900, David Hilbert put forth a list of the 23 most important unsolved problems in mathematics. At the head of the list was Cantor's continuum hypothesis. The problem was still open when the Liebers wrote their book. In 1966, a mathematician named Paul Cohen proved that the continuum hypothesis is actually independent of the generally accepted axioms of set theory, and earned the Fields medal for it.

    2 out of 2 people found this review helpful.

    Was this review helpful? Yes  No   Report this review
  • Posted February 18, 2011

    Infinity analyzed

    Nobody explains mathematical ideas for the layman as does Lillian R. Lieber. And the fanciful illustrations that always accompany her work, done by Hugh Gray Lieber, are amusing and informative. Infinity: Beyond the Beyond the Beyond presents an account of how mathematics has learned to deal with the infinite, primarily through the work of Georg Cantor. Controversial in its day, Cantor's set theory and transfinite arithmetic are now part of the foundations of modern mathematics. Perhaps the most startling idea to be had from this book is that infinite sets are not all of the same size. I have before me a copy of the 1953 original, as well as the 2007 abridgement. Aside from the fact that the older book is a hardcover, the abridgement is the better book. The editor, Barry Mazur, a mathematician at Harvard, has removed the dated, nonmathematical introductory material and the chapters on calculus. This book is now a superb layman's guide to the mathematics of transfinities. If you would like more biography and less mathematics, you might try The Mystery of the Aleph, by Amir D. Aczel. Note: In 1900, David Hilbert put forth a list of the 23 most important unsolved problems in mathematics. At the head of the list was Cantor's continuum hypothesis. The problem was still open when the Liebers wrote their book. In 1963, a mathematician named Paul Cohen proved that the continuum hypothesis is actually independent of the generally accepted axioms of set theory, and earned the Fields medal for it.

    1 out of 1 people found this review helpful.

    Was this review helpful? Yes  No   Report this review
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