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Logical Number Theory I

An Introduction

Craig Smorynski (Autor)

Buch | Softcover

X, 405 Seiten

1991 | 1. Softcover reprint of the original 1st ed. 1991
Springer Berlin (Verlag)
978-3-540-52236-2 (ISBN)

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Number theory as studied by the logician is the subject matter of the book. This first volume can stand on its own as a somewhat unorthodox introduction to mathematical logic for undergraduates, dealing with the usual introductory material: recursion theory, first-order logic, completeness, incompleteness, and undecidability. In addition, its second chapter contains the most complete logical discussion of Diophantine Decision Problems available anywhere, taking the reader right up to the frontiers of research (yet remaining accessible to the undergraduate). The first and third chapters also offer greater depth and breadth in logico-arithmetical matters than can be found in existing logic texts. Each chapter contains numerous exercises, historical and other comments aimed at developing the student's perspective on the subject, and a partially annotated bibliography.

I. Arithmetic Encoding.- 1. Polynomials.- 2. Sums of Powers.- 3. The Cantor Pairing function.- 4. The Fueter-Pólya Theorem, I.- *5. The Fueter-Pólya Theorem, II.- 6. The Chinese Remainder Theorem.- 7. The ?-Function and Other Encoding Schemes.- 8. Primitive Recursion.- *9. Ackermann Functions.- 10. Arithmetic Relations.- 11. Computability.- 12. Elementary Recursion Theory.- 13. The Arithmetic Hierarchy.- 14. Reading List.- II. Diophantine Encoding.- 1. Diophantine Equations; Some Background.- 2. Initial Results; The Davis-Putnam-Robinson Theorem.- 3. The Pell Equation, I.- 4. The Pell Equation, II.- 5. The Diophantine Nature of R.E. Relations.- 6. Applications.- 7. Forms.- *8. Binomial Coëfficients.- *9. A Direct Proof of the Davis-Putnam-Robinson Theorem.- *10. The 3-Variable Exponential Diophantine Result.- 11. Reading List.- III. Weak Formal Theories of Arithmetic.- 1. Ignorabimus?.- 2. Formal Language and Logic.- 3. The Completeness Theorem.- 4. Presburger-Skolem Arithmetic; The Theory of Addition.- *5. Skolem Arithmetic; The Theory of Multiplication.- 6. Theories with + and ?; Incompleteness and Undecidability.- 7. Semi-Repiesentability of Functions.- 8. Further Undecidability Results.- 9. Reading List.- Index of Names.- Index of Subjects.

Erscheint lt. Verlag	25.4.1991
Reihe/Serie	Universitext
Zusatzinfo	X, 405 p. 2 illus.
Verlagsort	Berlin
Sprache	englisch
Maße	170 x 242 mm
Gewicht	696 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Allgemeines / Lexika
	Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
	Mathematik / Informatik ► Mathematik ► Logik / Mengenlehre
	Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
Schlagworte	Computability Theory • diophantine equations • Diophantische Gleichung • HC/Mathematik/Grundlagen • incompleteness • Mathematical Logic • Number Theory • Proof • recursion theory • Rekursionsgleichung • Rekursionstheorie • undecidability • Unentscheidbarkeit • Unvollständigkeit • Zahlentheorie
ISBN-10	3-540-52236-0 / 3540522360
ISBN-13	978-3-540-52236-2 / 9783540522362
Zustand	Neuware