Undecidable Theories

Undecidable Theories : Studies in Logic and the Foundation of Mathematics

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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."show more

Product details

  • Paperback | 112 pages
  • 134.62 x 208.28 x 7.62mm | 136.08g
  • Dover Publications Inc.
  • New York, United States
  • English
  • Dover.
  • 0486477037
  • 9780486477039
  • 422,772

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 Indexshow more

About Alfred Tarski

Polish mathematician Alfred Tarski (1901-83) ranks among the greatest logicians of all time. Best known for his work on model theory, meta mathematics, and algebraic logic, he contributed to many other fields of mathematics and taught at the University of California, Berkeley, for more than 40 years.Tarski's student Andrzej Mostowksi worked at the University of Warsaw on first-order logic and model theory.Tarski's University of California colleague Raphael M. Robinson built on Tarski's concept of essential undecidability and proved a number of mathematical theories undecidable.show more