具体描述
Written by a noted expert on logic and set theory, this study of basic number systems explores natural numbers, integers, rational numbers, real numbers, and complex numbers. Geared toward undergraduate and beginning graduate students, it requires minimal mathematical training. Numerous exercises and several helpful appendixes supplement the text. 1973 edition.
《Number Systems and the Foundations of Analysis》是一本旨在深入探索数学分析领域核心概念和方法的著作。它系统地介绍了现代数学中数字系统的发展历程，并以严谨的逻辑结构呈现了复杂分析问题的解决路径。这本书不仅关注基础知识，更注重引导读者从实际案例中理解抽象理论，为进一步的研究提供坚实的基础。内容涵盖了数论、代数结构及其与分析方法的结合，帮助读者建立对数学系统运作原理的全面认识。书中详细讨论了集合论和模型理论在数学建模中的应用，通过具体例子展示了如何用严密的思维处理复杂问题。在探讨实数、复数以及更高维度数字系统的过程中，作者特别强调这些知识点对于理解现代分析工具的重要性。此外，该书还深入剖析了递归关系和函数的构造方法，使读者能够从理论出发，掌握如何在不同情境下应用分析技术。内容丰富且结构清晰，为初学者和进阶学习者提供了全面而深入的思考平台。书中对历史发展路径的剖析还赋予了其独特的学术价值，帮助读者更好地理解数学思想的演变与迁移。在实践层面，该书通过丰富的例题和问题解答，强化了读者对理论知识的掌握和实际运用能力。整体而言，这本书不仅是一篇系统性的学习指南，更是一个深入探讨数学思维方式的典范。它以严谨的逻辑与清晰的语言，为读者打开了一扇理解复杂数学世界的大门，启示着更深层次的思考和创新应用。 该书采用详细而有条理的写作风格，逐步引导读者理解每一部分内容，并结合实际情境进行分析，使知识点更易于传达。这种结构不仅帮助初学者明确学习重点，也为资深读者提供了拓展思维、激发研究的契机。整个内容没有预设任何特定观点，而是以客观、全面的态度呈现数学系统的本质与发展脉络，充分体现了对学术研究的敬重和细致。通过对各类数学理论的系统阐述，该书为读者打下坚实的理论基础，同时提升他们解决复杂数学问题的能力。总体而言，这是一份兼具教育性与参考性的经典作品，适合希望深入探索分析学科内涵的读者。
作者简介
目录信息
Chapter 1. Basic Facts and Notions of Logic and Set Theory
1.1 Logical Connectives
1.2 Conditionals
1.3 Biconditionals
1.4 Quantifiers
1.5 Sets
1.6 Membership. Equality and Inclusion of Sets
1.7 The Empty Set
1.8 Union and Intersection
1.9 Difference and Complement
1.10 Power Set
1.11 Arbitrary Unions and Intersections
1.12 Ordered Pairs
1.13 Cartesian Product
1.14 Relations
1.15 Inverse and Composition of Relations
1.16 Reflexivity, Symmetry, and Transitivity
1.17 Equivalence Relations
1.18 Functions
1.19 Functions from A into (Onto) B
1.20 One-One Functions
1.21 Composition of Functions
1.22 Operations
Chapter 2. The Natural Numbers
2.1 Peano Systems
2.2 The Iteration Theorem
2.3 Application of the Iteration Theorem: Addition
2.4 The Order Relation
2.5 Multiplication
2.6 Exponentiation
2.7 Isomorphism, Categoricity
2.8 A Basic Existence Assumption
Supplementary Exercises
Suggestions for Further Reading
Chapter 3. The Integers
3.1 Definition of the Integers
3.2 Addition and Multiplication of Integers
3.3 Rings and Integral Domains
3.4 Ordered Integral Domains
3.5 Greatest Common Divisor, Primes
3.6 Integers Modulo n
3.7 Characteristic of an Integral Domain
3.8 Natural Numbers and Integers of an Integral Domain
3.9 Subdomains, Isomorphisms, Characterizations of the Integers
Supplementary Exercises
Chapter 4. Rational Numbers and Ordered Fields
4.1 Rational Numbers
4.2 Fields
4.3 Quotient field of an Integral Domain
4.4 Ordered Fields
4.5 Subfields. Rational Numbers of a Field.
Chapter 5. The Real Number System
5.1 Inadequacy of the Rationals
5.2 Archimedean Ordered Fields
5.3 Least Upper Bounds and Greatest Lower Bounds
5.4 The Categoricity of the Theory of Complete Ordered Fields
5.5 Convergent Sequences and Cauchy Sequences
5.6 Cauchy Completion. The Real Number System
5.7 Elementary Topology of the Real Number System
5.8 Continuous Functions
5.9 Infinite Series
Appendix A. Equality
Appendix B. Finite Sums and the Sum Notation
Appendix C. Polynomials
Appendix D. Finite, Infinite, and Denumerable Sets. Cardinal Numbers
Appendix E. Axiomatic Set Theory and the Existence of a Peano System
Appendix F. Construction of the Real Numbers via Dedekind Cuts
Appendix G.Complex Numbers
Bibliography
Index of Special Symbols
Index
· · · · · · (收起)
1.1 Logical Connectives
1.2 Conditionals
1.3 Biconditionals
1.4 Quantifiers
1.5 Sets
1.6 Membership. Equality and Inclusion of Sets
1.7 The Empty Set
1.8 Union and Intersection
1.9 Difference and Complement
1.10 Power Set
1.11 Arbitrary Unions and Intersections
1.12 Ordered Pairs
1.13 Cartesian Product
1.14 Relations
1.15 Inverse and Composition of Relations
1.16 Reflexivity, Symmetry, and Transitivity
1.17 Equivalence Relations
1.18 Functions
1.19 Functions from A into (Onto) B
1.20 One-One Functions
1.21 Composition of Functions
1.22 Operations
Chapter 2. The Natural Numbers
2.1 Peano Systems
2.2 The Iteration Theorem
2.3 Application of the Iteration Theorem: Addition
2.4 The Order Relation
2.5 Multiplication
2.6 Exponentiation
2.7 Isomorphism, Categoricity
2.8 A Basic Existence Assumption
Supplementary Exercises
Suggestions for Further Reading
Chapter 3. The Integers
3.1 Definition of the Integers
3.2 Addition and Multiplication of Integers
3.3 Rings and Integral Domains
3.4 Ordered Integral Domains
3.5 Greatest Common Divisor, Primes
3.6 Integers Modulo n
3.7 Characteristic of an Integral Domain
3.8 Natural Numbers and Integers of an Integral Domain
3.9 Subdomains, Isomorphisms, Characterizations of the Integers
Supplementary Exercises
Chapter 4. Rational Numbers and Ordered Fields
4.1 Rational Numbers
4.2 Fields
4.3 Quotient field of an Integral Domain
4.4 Ordered Fields
4.5 Subfields. Rational Numbers of a Field.
Chapter 5. The Real Number System
5.1 Inadequacy of the Rationals
5.2 Archimedean Ordered Fields
5.3 Least Upper Bounds and Greatest Lower Bounds
5.4 The Categoricity of the Theory of Complete Ordered Fields
5.5 Convergent Sequences and Cauchy Sequences
5.6 Cauchy Completion. The Real Number System
5.7 Elementary Topology of the Real Number System
5.8 Continuous Functions
5.9 Infinite Series
Appendix A. Equality
Appendix B. Finite Sums and the Sum Notation
Appendix C. Polynomials
Appendix D. Finite, Infinite, and Denumerable Sets. Cardinal Numbers
Appendix E. Axiomatic Set Theory and the Existence of a Peano System
Appendix F. Construction of the Real Numbers via Dedekind Cuts
Appendix G.Complex Numbers
Bibliography
Index of Special Symbols
Index
· · · · · · (收起)
读后感
用户评价
课程讲义http://www.mas.ncl.ac.uk/~najd2/teaching/mas121/#tutorials，但不是本书
课程讲义http://www.mas.ncl.ac.uk/~najd2/teaching/mas121/#tutorials，但不是本书
课程讲义http://www.mas.ncl.ac.uk/~najd2/teaching/mas121/#tutorials，但不是本书
搭配作者的另一本数理逻辑一起看的。终于把实数的建立和各种性质搞清楚了！
课程讲义http://www.mas.ncl.ac.uk/~najd2/teaching/mas121/#tutorials，但不是本书
