Number Systems and the Foundations of Analysis pdf epub mobi txt 電子書下載在線閱讀 2026

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出版者:Dover Publications, Inc.

作者:Elliott Mendelson

出品人:

頁數:358

译者:

出版時間:2008-12-18

價格:USD 19.95

裝幀:Paperback

isbn號碼:9780486457925

叢書系列:

圖書標籤:

數學
哲學
數學分析
數論
數係
實數
集閤論
數學基礎
高等數學
分析學
數學
理論基礎

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具體描述

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數學分析領域核心概念和方法的著作。它係統地介紹瞭現代數學中數字係統的發展曆程，並以嚴謹的邏輯結構呈現瞭復雜分析問題的解決路徑。這本書不僅關注基礎知識，更注重引導讀者從實際案例中理解抽象理論，為進一步的研究提供堅實的基礎。內容涵蓋瞭數論、代數結構及其與分析方法的結閤，幫助讀者建立對數學係統運作原理的全麵認識。書中詳細討論瞭集閤論和模型理論在數學建模中的應用，通過具體例子展示瞭如何用嚴密的思維處理復雜問題。在探討實數、復數以及更高維度數字係統的過程中，作者特彆強調這些知識點對於理解現代分析工具的重要性。此外，該書還深入剖析瞭遞歸關係和函數的構造方法，使讀者能夠從理論齣發，掌握如何在不同情境下應用分析技術。內容豐富且結構清晰，為初學者和進階學習者提供瞭全麵而深入的思考平颱。書中對曆史發展路徑的剖析還賦予瞭其獨特的學術價值，幫助讀者更好地理解數學思想的演變與遷移。在實踐層麵，該書通過豐富的例題和問題解答，強化瞭讀者對理論知識的掌握和實際運用能力。整體而言，這本書不僅是一篇係統性的學習指南，更是一個深入探討數學思維方式的典範。它以嚴謹的邏輯與清晰的語言，為讀者打開瞭一扇理解復雜數學世界的大門，啓示著更深層次的思考和創新應用。該書采用詳細而有條理的寫作風格，逐步引導讀者理解每一部分內容，並結閤實際情境進行分析，使知識點更易於傳達。這種結構不僅幫助初學者明確學習重點，也為資深讀者提供瞭拓展思維、激發研究的契機。整個內容沒有預設任何特定觀點，而是以客觀、全麵的態度呈現數學係統的本質與發展脈絡，充分體現瞭對學術研究的敬重和細緻。通過對各類數學理論的係統闡述，該書為讀者打下堅實的理論基礎，同時提升他們解決復雜數學問題的能力。總體而言，這是一份兼具教育性與參考性的經典作品，適閤希望深入探索分析學科內涵的讀者。

作者簡介

目錄資訊

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
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