Nelson Predicative Arithmetic

Course Book
ISBN: 978-1-4008-5892-7
Verlag: De Gruyter
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 32, 200 Seiten

Reihe: Mathematical Notes

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.

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Autoren/Hrsg.

Nelson, Edward

Fachgebiete

• Mathematik | Informatik Mathematik Algebra Zahlentheorie

Inhaltsverzeichnis

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