Automated Development of Fundamental Mathematical Theories

1993
ISBN: 978-0-7923-2021-0
Verlag: Springer Netherlands

Buch, Englisch, 273 Seiten, Format (B × H): 160 mm x 241 mm, Gewicht: 1310 g

Reihe: Automated Reasoning Series

The author provides an introduction to automated reasoning, and in particular to resolution theorem proving using the prover OTTER. He presents a new clausal version of von Neumann-Bernays-Gödel set theory, and lists over 400 theorems proved semiautomatically in elementary set theory. He presents a semiautomated proof that the composition of homomorphisms is a homomorphism, thus solving a challenge problem.
The author next develops Peano's arithmetic, and gives more than 1200 definitions and theorems in elementary number theory. He gives part of the proof of the fundamental theorem of arithmetic (unique factorization), and gives and OTTER-generated proof of Euler's generalization of Fermat's theorem.
Next he develops Tarski's geometry within OTTER. He obtains proofs of most of the challenge problems appearing in the literature, and offers further challenges. He then formalizes the modal logic calculus K4, in order to obtain very high level automated proofs of Löb's theorem, and of Gödel's two incompleteness theorems. Finally he offers thirty-one unsolved problems in elementary number theory as challenge problems.

Zielgruppe

Research

Autoren/Hrsg.

Quaife, Art

Fachgebiete

• Mathematik | Informatik Mathematik Mathematik Allgemein Grundlagen der Mathematik
• Technische Wissenschaften Elektronik | Nachrichtentechnik Elektronik Robotik
• Mathematik | Informatik Mathematik Mathematik Allgemein Mathematische Logik
• Mathematik | Informatik EDV | Informatik Informatik Künstliche Intelligenz Wissensbasierte Systeme, Expertensysteme

Inhaltsverzeichnis

Preface: A Personal View of Automated Reasoning Research. 1. Introduction to Automated Reasoning. 2. Von Neumann-Bernays-Gödel Set Theory. 3. Peano's Arithmetic. 4. Tarski's Geometry. 5. Löb's Theorem and Gödel's Two Incompleteness Theorems. 6. Unsolved Problems in Elementary Number Theory. Appendix 1: Gödel's Axioms for Set Theory. Appendix 2: Theorems Proved in NBG Set Theory. Appendix 3: Theorems Proved in Peano's Arithmetic. Bibliography. Index of Names. Index of Subjects.