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Full Description
This book develops arithmetic without the induction principle, working in theories that are interpretable in Raphael Robinson's theory Q. Certain inductive formulas, the bounded ones, are interpretable in Q. A mathematically strong, but logically very weak, predicative arithmetic is constructed. Originally published in 1986. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.
Contents
*FrontMatter, pg. i*Acknowledgments, pg. v*Table of Contents, pg. vii*Chapter 1. The impredicativity of induction, pg. 1*Chapter 2. Logical terminology, pg. 3*Chapter 3. The axioms of arithmetic, pg. 8*Chapter 4. Order, pg. 10*Chapter 5. Induction by relativization, pg. 12*Chapter 6. Interpretability in Robinson's theory, pg. 16*Chapter 7. Bounded induction, pg. 23*Chapter 8. The bounded least number principle, pg. 29*Chapter 9. The euclidean algorithm, pg. 32*Chapter 10. Encoding, pg. 36*Chapter 11. Bounded separation and minimum, pg. 43*Chapter 12. Sets and functions, pg. 46*Chapter 13. Exponential functions, pg. 51*Chapter 14. Exponentiation, pg. 54*Chapter 15. A stronger relativization scheme, pg. 60*Chapter 16. Bounds on exponential functions, pg. 64*Chapter 17. Bounded replacement, pg. 70*Chapter 18. An impassable barrier, pg. 73*Chapter 19. Sequences, pg. 82*Chapter 20. Cardinality, pg. 90*Chapter 21. Existence of sets, pg. 95*Chapter 22. Semibounded Replacement, pg. 98*Chapter 23. Formulas, pg. 101*Chapter 24. Proofs, pg. 111*Chapter 25. Derived rules of inference, pg. 115*Chapter 26. Special constants, pg. 134*Chapter 27. Extensions by definition, pg. 136*Chapter 28. Interpretations, pg. 152*Chapter 29. The arithmetization of arithmetic, pg. 157*Chapter 30. The consistency theorem, pg. 162*Chapter 31. Is exponentiation total?, pg. 173*Chapter 32. A modified Hilbert program, pg. 178*Bibliography, pg. 181*General index, pg. 183*Index of defining axioms, pg. 186
