In the Light of Logic

Hardcover | October 1, 1998

bySolomon Feferman

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In this collection of essays written over a period of twenty years, Solomon Feferman explains advanced results in modern logic and employs them to cast light on significant problems in the foundations of mathematics. Most troubling among these is the revolutionary way in which Georg Cantorelaborated the nature of the infinite, and in doing so helped transform the face of twentieth-century mathematics. Feferman details the development of Cantorian concepts and the foundational difficulties they engendered. He argues that the freedom provided by Cantorian set theory was purchased at aheavy philosophical price, namely adherence to a form of mathematical platonism that is difficult to support. Beginning with a previously unpublished lecture for a general audience, Deciding the Undecidable, Feferman examines the famous list of twenty-three mathematical problems posed by David Hilbert, concentrating on three problems that have most to do with logic. Other chapters are devoted to the workand thought of Kurt Godel, whose stunning results in the 1930s on the incompleteness of formal systems and the consistency of Cantors continuum hypothesis have been of utmost importance to all subsequent work in logic. Though Godel has been identified as the leading defender of set-theoreticalplatonism, surprisingly even he at one point regarded it as unacceptable. In his concluding chapters, Feferman uses tools from the special part of logic called proof theory to explain how the vast part--if not all--of scientifically applicable mathematics can be justified on the basis of purely arithmetical principles. At least to that extent, the question raised in twoof the essays of the volume, Is Cantor Necessary?, is answered with a resounding no. This volume of important and influential work by one of the leading figures in logic and the foundations of mathematics is essential reading for anyone interested in these subjects.

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In this collection of essays written over a period of twenty years, Solomon Feferman explains advanced results in modern logic and employs them to cast light on significant problems in the foundations of mathematics. Most troubling among these is the revolutionary way in which Georg Cantorelaborated the nature of the infinite, and in d...

Solomon Feferman is at Stanford University.

other books by Solomon Feferman

Alfred Tarski: Life and Logic
Alfred Tarski: Life and Logic

Paperback|Apr 7 2008

$29.95

Format:HardcoverDimensions:352 pages, 6.3 × 9.29 × 1.3 inPublished:October 1, 1998Publisher:Oxford University Press

The following ISBNs are associated with this title:

ISBN - 10:0195080300

ISBN - 13:9780195080308

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Table of Contents

I: Foundational Problems1. Deciding the undecidable: Wrestling with Hilbert's Problems2. Infinity in mathematics: Is Cantor necessary?II: Foundational Ways3. The logic of mathematical discovery vs. the logical structure of mathematics4. Foundational Ways5. Working FoundationsIII: Godel6. Godel's life and work7. Kurt Godel: conviction and caution8. Introductory note to Godel's 1933 lectureIV: Proof Theory9. What does logic have to tell us about mathematical proofs?10. What rests on what? The proof-theoretic analysis of mathematics11. Godel's Dialectica interpretation and its two-way stretchV: Countably Reducible Mathematics12. Infinity in mathematics: Is Cantor necessary? (Conclusion)13. Weyl vindicated: Das Kontinuum 70 years later14. Why a little bit goes a long way: Logical Foundations of scientifically applicable mathematics

Editorial Reviews

"...the papers in this book provide an illuminating picture of much of the past work and a good deal of current progress in the foundations of mathematics. The author, one of the most distinguished contributors to that progress within the last four decades, is an excellent expositor of thevarious issues in the forefront of recent research..."--Mathematical Reviews