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Kurt Godel, the greatest logician of our time, startled the world of mathematics in 1931 with his Theorem of Undecidability, which showed that some statements in mathematics are inherently "undecidable." His work on the completeness of logic, the incompleteness of number theory, and the consistency of the axiom of choice and the continuum theory brought him further worldwide fame. In this introductory volume, Raymond Smullyan, himself a well-known logician, guides the reader through the fascinating world of Godel's incompleteness theorems. The level of presentation is suitable for anyone with a basic acquaintance with mathematical logic. As a clear, concise introduction to a difficult but essential subject, the book will appeal to mathematicians, philosophers, and computer scientists.
Table of Contents
1. The General Idea Behind Gdel's Proof
2. Tarski's Theorem for Arithmetic
3. The Incompleteness of Peano Arithmetic with Exponentation
4. Arithmetic Without the Exponential
5. Gdel's Proof Based on Consistency
6. Rosser Systems
7. Shepherdson's Representation Theorems
8. Definability and Diagonalization
9. The Unprovability of Consistency
10. Some General Remarks on Provability and Truth
11. Self-Referential Systems