數(shù)理邏輯

內(nèi)容概要

　　What is a mathematical proof? How can proofs be justified? Are there limitations to provability? To what extent can machines carry out mathematical proofs?　　Only in this century has there been success in obtaining substantial and satisfactory answers. The present book contains a systematic discussion of these results. The investigations are centered around first-order logic. Our first goal is Godels completeness theorem， which shows that the consequence relation coincides with formal provability： By means of a calculus consisting of simple formal inference rules， one can obtain all consequences of a given axiom system (and in particular， imitate all mathematical proofs)

作者簡介

作者：(德國)艾賓浩斯(Ebbinghaus H.D.)

書籍目錄

Preface PART A I Introduction 1.An Example from Group Theory 2.An Example from the Theory of Equivalence Relations 3.A Preliminary Analysis 4.Preview II Syntax of First-Order Languages 1.Alphabets 2.The Alphabet of a First-Order Language 3.Terms and Formulas in First-Order Languages 4.Induction in the Calculus of Terms and in the Calculus of Formulas 5.Free Variables and Sentences III Semantics of First-Order Languages 1.Structures and Interpretations 2.Standardization of Connectives 3.The Satisfaction Relation 4.The Consequence Relation 5.Two Lemmas on the Satisfaction Relation ……

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《數(shù)理邏輯(第2版)》由世界圖書出版公司出版。A short digression into model theory will help us to analyze the expressive power of the first-order language， and it will turn out that there are certain deficiencies. For example， the first-order language does not allow the formulation of an adequate axiom system for arithmetic or analysis. On the other hand， this di~~culty can be overcome——-even in the framework of first-order logic——by developing mathematics in set-theoretic terms. We explain the prerequisites from set theory necessary for this purpose and then treat the subtle relation between logic and set theory in a thorough manner.Godel‘s incompleteness theorems are presented in connection with several related results （such as Trahtenbrot’s theorem） which all exemplify the limitatious of machine-oriented proof methods. The notions of computability theory that are relevant to this discussion are given in detail.

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