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Diagonalization and self-reference / Raymond M. Smullyan

By: Smullyan, Raymond M.
Series: Oxford logic guides ; 27. Publisher: Oxford, U.K. : Clarendon Press, 1994Description: xv, 396 p. ; 24 cm.ISBN: 0198534507.Subject(s): Fixed point theory | Recursion theoryDDC classification: 511.3
Contents:
1. Introduction to Self-Reference and Recursion. - 1. Introduction to self-reference. 2. Some classical fixed point arguments compared. 3. How to silence a universal machine. 4. Some general incompleteness theorems. 5. Self-reference in arithmetic. 6. Introduction to formal systems and recursion. 7. A universal system and its applications. - II. Systems with Effective Properties. 8. Arithmetization of formal systems. 9. Elementary formal systems and incompleteness proofs. 10. Doubly indexed relational systems. 11. Effective representation systems. - III. Fixed Point Theorems in a General Setting. 12. Sequential systems. 13. Strong fixed point properties. 14. Multiple fixed point properties. 15. Synchronization and pairing functions. 16. Some further relations between fixed point properties. - IV. Combinators and Sequential Systems. 17. Fixed point properties of combinatory logic. 18. Formal combinatory logic. 19. A second variety of fixed point theorems. 20. Extended sequential systems. - References. - Index.
Summary: This volume is written for the beginner and expert alike, since it is both an introduction to self-reference, diagonalization, and fixed points, as they occur in Godel's incompleteness proofs, recursion theory, combinatory logic, semantics, and metamathematics, and a presentation of new results - partly in these areas, but mostly in their synthesis, as developed in the last nine chapters. No prior knowledge of these fields is presupposed, except that for the applications to mathematical logic in Chapter 5 and 9, a little acquaintance with the logical connectives and quantifiers, though not strictly necessary, is desirable. - Preface.
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511.3 SMU (Browse shelf) 1 Available GENxx,GENxx,02,GR 5000083217 Please fill up online form at https://taylorslibrary.taylors.edu.my/services/external_storage1

1. Introduction to Self-Reference and Recursion. - 1. Introduction to self-reference. 2. Some classical fixed point arguments compared. 3. How to silence a universal machine. 4. Some general incompleteness theorems. 5. Self-reference in arithmetic. 6. Introduction to formal systems and recursion. 7. A universal system and its applications. - II. Systems with Effective Properties. 8. Arithmetization of formal systems. 9. Elementary formal systems and incompleteness proofs. 10. Doubly indexed relational systems. 11. Effective representation systems. - III. Fixed Point Theorems in a General Setting. 12. Sequential systems. 13. Strong fixed point properties. 14. Multiple fixed point properties. 15. Synchronization and pairing functions. 16. Some further relations between fixed point properties. - IV. Combinators and Sequential Systems. 17. Fixed point properties of combinatory logic. 18. Formal combinatory logic. 19. A second variety of fixed point theorems. 20. Extended sequential systems. - References. - Index.

This volume is written for the beginner and expert alike, since it is both an introduction to self-reference, diagonalization, and fixed points, as they occur in Godel's incompleteness proofs, recursion theory, combinatory logic, semantics, and metamathematics, and a presentation of new results - partly in these areas, but mostly in their synthesis, as developed in the last nine chapters. No prior knowledge of these fields is presupposed, except that for the applications to mathematical logic in Chapter 5 and 9, a little acquaintance with the logical connectives and quantifiers, though not strictly necessary, is desirable. - Preface.