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　　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.
　　Godels incompleteness theorems are presented in connection with several related results (such as Trahtenbrots 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. The concept of computability is made precise by means of the register machine as a

内容简介

　　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)

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目录

Preface
PART A
ⅠIntroduction
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
6.Some simple formalizations
7.Some remarks on Formalizability
8.Substitution
Ⅳ A Sequent Calculus
1．Sequent Rules
2．Structural Rules and Connective Rules
3．Derivable Connective Rules
4．Quantifier and Equality Rules
5．Further Derivable Rules and Sequents
6．Summary and Example
7．Consistency
ⅤThe Completeness Theorem
1．Henkin’S Theorem．
2． Satisfiability of Consistent Sets of Formulas（the Countable Casel
3． Satisfiability of Consistent Sets of Formulas（the General Case）
4．The Completeness Theorem
Ⅵ The LSwenheim-Skolem and the Compactness Theorem
1．The L6wenheim-Skolem Theorem．
2．The Compactness Theorem
3．Elementary Classes
4．Elementarily Equivalent Structures
Ⅶ The Scope of First-Order Logic
1．The Notion of Formal Proof
2．Mathematics Within the Framework of Fimt—Order Logic
3．The Zermelo-Fraenkel Axioms for Set Theory．
4．Set Theory as a Basis for Mathematics
Ⅷ Syntactic Interpretations and Normal Forms
1．Term-Reduced Formulas and Relational Symbol Sets
2．Syntactic Interpretations
3．Extensions by Definitions
4．Normal Forms
PART B
Ⅸ Extensions of First-order logic
Ⅹ Limitations of the Formal Method
Ⅺ Free Models and Logic Programming
Ⅻ An Algebraic Characterization of Elementary Equivalence
ⅩⅢ Lindstrom’s Theorems
References
Symbol Index
Subject Index

前言/序言

数理逻辑（第2版） [Mathematical Logic] 电子书 下载 mobi epub pdf txt

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逻辑问题，初步奠定了数理逻辑的基础。

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基础数学

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所谓数学方法就是指数学采用的一般方法，包括使用符号和公式，已有的数学成果和方法，特别是使用形式的公理方法。

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编辑本段

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好书

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编辑本段

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产生

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命题演算是研究关于命题如何通过一些逻辑连接词构成更复杂的命题以及逻辑推理的方法。命题是指具有具体意义的又能判断它是真还是假的句子。 如果我们把命题看作运算的对象，如同代数中的数字、字母或代数式，而把逻辑连接词看作运算符号，就象代数中的“加、减、乘、除”那样，那么由简单命题组成复和命题的过程，就可以当作逻辑运算的过程，也就是命题的演算。

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正版

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