The Logic of Provability
Cambridge University Press
Edition: New Ed, 8/21/2008
EAN 9780521483254, ISBN10: 0521483255
Paperback, 316 pages, 22.9 x 15.2 x 2 cm
Language: English
Originally published in English
This book, written by one of the most distinguished of contemporary philosophers of mathematics, is a fully rewritten and updated successor to the author's earlier The Unprovability of Consistency (1979). Its subject is the relation between provability and modal logic, a branch of logic invented by Aristotle but much disparaged by philosophers and virtually ignored by mathematicians. Here it receives its first scientific application since its invention. Modal logic is concerned with the notions of necessity and possibility. What George Boolos does is to show how the concepts, techniques, and methods of modal logic shed brilliant light on the most important logical discovery of the twentieth century: the incompleteness theorems of Kurt Godel and the 'self-referential' sentences constructed in their proof. The book explores the effects of reinterpreting the notions of necessity and possibility to mean provability and consistency.
1. GL and other systems of propositional modal logic
2. Peano arithmetic
3. The box as Bew(x)
4. Semantics for GL and other modal logics
5. Completeness and decidability of GL and K, K4, T, B, S4, and S5
6. Canonical models
7. On GL
8. The fixed point theorem
9. The arithmetical completeness theorems for GL and GLS
10. Trees for GL
11. An incomplete system of modal logic
12. An S4 -preserving proof-theoretical treatment of modality
13. Modal logic within set theory
14. Modal logic within analysis
15. The joint provability logic of consistency and w-consistency
16. On GLB
the fixed point theorem, letterless sentences, and analysis
18. Quantified provability logic with one one-place predicate letter
Notes
Bibliography
Index.