Undecidable Theories: Studies in Logic and the Foundation of Mathematics

Undecidable Theories: Studies in Logic and the Foundation of Mathematics

Paperback Dover Books on Mathematics

By (author) Alfred Tarski, With Andrzej Mostowski, With Raphael R. Robinson

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  • Publisher: Dover Publications Inc.
  • Format: Paperback | 112 pages
  • Dimensions: 135mm x 208mm x 8mm | 136g
  • Publication date: 24 September 2010
  • Publication City/Country: New York
  • ISBN 10: 0486477037
  • ISBN 13: 9780486477039
  • Edition statement: Dover.
  • Sales rank: 243,681

Product description

This graduate-level book is well known for its proof that many mathematical systems--including lattice theory, abstract projective geometry, and closure algebras--are undecidable. Based on research conducted from 1938 to 1952, it consists of three treatises by a prolific author who ranks among the greatest logicians of all time. The first article, "A General Method in Proofs of Undecidability," examines theories with standard formalization, undecidable theories, interpretability, and relativization of quantifiers. The second feature, "Undecidability and Essential Undecidability in Mathematics," explores definability in arbitrary theories and the formalized arithmetic of natural numbers. It also considers recursiveness, definability, and undecidability in subtheories of arithmetic as well as the extension of results to other arithmetical theories. The compilation concludes with "Undecidability of the Elementary Theory of Groups."

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Table of contents

I.  A General Method in Proofs of Undecidability by Alfred Tarski  I.1.  Introduction  I.2.  Theories with standard formalization  I.3.  Undecidable and essentially undecidable theories  I.4.  Interpretability and weak interpretability  I.5.  Relativization of quantifiers  I.6.  Examples and applications II.  Undecidability and Essential Undecidability in Arithmetic by Andrzej Mostowski, Raphael M. Robinson, and Alfred Tarski  II.1.  A summary of results; notation  II.2.  Definability in arbitrary theories  II.3.  Formalized arithmetic of natural numbers and its subtheories  II.4.  Recursiveness and definability in subtheories of arithmetic  II.5.  Undecidability of subtheories of arithmetic  II.6.  Extension of the results to other arithmetical theories and to various theories of rings III.  Undecidability of the Elementary Theory of Groups by Alfred Tarski Bibliography Index