《集合论:对无穷概念的探索》是“逻辑与形而上学教科书系列”中的一本，书中介绍了集合论的基础知识，共有集合与公理，关系与函数，实数的构造，基数，滤、理想与无界闭集，集合的宇宙，可构成集，力迫等9章内容；除了讨论集合论的基本概念，还讨论了可构成集、力迫法等现代内容，同时还讨论了与连续统假设相关的一些哲学问题。

本卷是这本《集合论导引》的开卷，分为三章，是后续两卷的基础。第1章主要是引进集合论的基本公理、基本概念、基本方法，并给出典型的可数集合的例子，包括自然数集合、整数集合、有理数集合以及彻底有限集合等。第2章主要是引进选择公理以及由此建立起来的基数运算律和一些典型组合实例。第3章专门引进实数集合并对它进行系统分析。本卷将建立一系列基本概念，为全书作铺垫。

格奥尔格·康托（1845—1918），德国数学家，集合论的创始人。本书是一部数学经典。它记录了一百年前数学领域的一项惊人成就，也是数学和哲学思想史上关于无穷观念的一场革命。康托完全背离了自古希腊以来千余年的数学传统，创立了集合理论，提出了超穷序数和超穷基数理论，第一次使人们相信，自然数集合与有理数集合是可数的，而实数集合是不可数的；也第一次使人们相信，无穷不仅是存在的，无穷还可以比较大小，甚至无穷可以进行超穷的运算。他所创立的无穷理论，不仅直接促使现代集合论的建立，也极大地推进了数理逻辑的发展，而逻辑和现代集合论构成了全部数学的基础。

Set theory is concerned with the foundation of mathematics. In the original formulations of set theory, there were paradoxes contained in the idea of the "set of all sets". Current standard theory (Zermelo-Fraenkel) avoids these paradoxes by restricting the way sets may be formed by other sets, specifically to disallow the possibility of forming the set of all sets. In the 1930s, Quine proposed a different form of set theory in which the set of all sets - the universal set - is allowed, but other restrictions are placed on these axioms. Since then, the steady interest expressed in these non-standard set theories has been boosted by their relevance to computer science. The second edition still concentrates largely on Quine's New Foundations, reflecting the author's belief that this provides the richest and most mysterious of the various systems dealing with set theories with a universal set. Also included is an expanded and completely revised account of the set theories of Church-Oswald and Mitchell, with descriptions of permutation models and extensions that preserve power sets. Dr Foster here presents the reader with a useful and readable introduction for those interested in this topic, and a reference work for those already involved in this area.

This is an extensively revised edition of Mr. Quine's introduction to abstract set theory and to various axiomatic systematizations of the subject. The treatment of ordinal numbers has been strengthened and much simplified, especially in the theory of transfinite recursions, by adding an axiom and reworking the proofs. Infinite cardinals are treated anew in clearer and fuller terms than before. Improvements have been made all through the book; in various instances a proof has been shortened, a theorem strengthened, a space-saving lemma inserted, an obscurity clarified, an error corrected, a historical omission supplied, or a new event noted.

现代逻辑起源于19与20世纪之交人类对于数学基础问题的思考。其早期发展无疑是对人类理性事业的重大推动。弗雷格概念文字、希尔伯特公理系统、塔斯基真定义、哥德尔完备性与不完备性定理、图灵机对机械可计算的刻画等成果不仅让人类对数学基础有了全新的理解，孕育了现代计算机科学，还促成了分析哲学的崛起，由此深刻地改变了哲学研究的样貌。 此后的数理逻辑越来越陷入高度的技术化与专业化，并渐渐淡出公众、哲学家甚至其他领域数学家的视野。然而，无论是数学基础问题还是一般的哲学问题都仍然呼唤着哲学与逻辑的再度携手。杨睿之*的《作为哲学的数理逻辑/逻辑与形而上学教科书系列》试图将经过演变与发展的数理逻辑与哲学重新介绍给对方，让哲学工作者或爱好者快速正确地理解当代数理逻辑的主要工作，也让逻辑学工作者或学生了解与他们的工作有关的哲学上的关切。

For 30 years, Quine, a dominant figure in logical theory and philosophy of logic, has combined insights in methodology, language, epistemology, and ontology, to blur the boundaries of speculative metaphysics and natural sciences. This revised text contains two new essays with replies from Quine.

Hailed by the "Bulletin of the American Mathematical Society" as "undoubtedly a major addition to the literature of mathematical logic," this volume examines the essential topics and theorems of mathematical reasoning. No background in logic is assumed, and the examples are chosen from a variety of mathematical fields. 1978 edition.

These nine essays are largely concerned with the theory of meaning and references—semantics. At the same time adjacent portions of phil osophy and logic are discussed. To the existence of what objects may a given scientific theory be said to be committed? And what considerations may suitably guide us in accepting or revising such ontological commitments? These are among the questions dealt with in this book, particular attention being devoted to the role of abstract entities in mathematics. There is speculation on the mechanism whereby objects of one sort or another come to be posited a process in which the notion of identity plays an important part.

内容提要 本卷包括两部论著：《威拉德·范·奥曼·蒯因自传》（1986）和《数理逻辑》（1940）。前者是蒯因为“在世哲学家文库”《蒯因哲学》卷所写的简明自传；后者是蒯因的一部重要的逻辑著作，在其中，他仍然坚持逻辑主义纲领，试图从逻辑中推导出数学，把先前在《数理逻辑的新基础》（1937）一文中提出的NF系统，修改为ML系统，这是一个把命题逻辑、量化理论和集合论集为一身的系统，结构简明、特性奇异。 目录 威拉德·范·奥曼·蒯因自传 数理逻辑 导言 第一章 句子 1 合取、析取和否定 2 条件句 3 多重复合 4 使用和谈论 5 谈论句子的句子 6 准引语 7 括号与黑点 8 归约为三个初始联结词 9 归约为一个初始联结词 10 重言式 11 某些重言的形式 第二章 量化 12 量词 13 公式 14 约束、自由、闭包 15 量化的公理 16 定理 17 元定理 18 等值代换 19 存在量词 20 量词的分配 21 字母变体 第三章 项 22 类和分子 23 逻辑公式 24 抽离 25 等同 26 再论抽离 27 摹状词和名字 第四章 关于类的进一步理论 28 层次 29 更进一步的从属关系公理 30 等同可代入性 31 变元的代入 32 进一步的结果 33 逻辑积、和、补 34 包含 35 单类 第五章 关系 36 对和关系 37 关系的抽离 38 逆、象、关系积 39 祖先 40 函数 41 函数的抽离 42 作为关系的等同和从属 第六章 数 43 零、一、后继 44 自然数 45 可数集合 46 有穷的和无穷的 47 关系的幂 48 算术和、积、幂 …… 第七章 句法 附录 参考文献 人名索引 主题词索引

This is an introductory undergraduate textbook in set theory. In mathematics these days, essentially everything is a set. Some knowledge of set theory is necessary part of the background everyone needs for further study of mathematics. It is also possible to study set theory for its own interest--it is a subject with intruiging results anout simple objects. This book starts with material that nobody can do without. There is no end to what can be learned of set theory, but here is a beginning.

The main notions of set theory (cardinals, ordinals, transfinite induction) are fundamental to all mathematicians, not only to those who specialize in mathematical logic or set-theoretic topology. Basic set theory is generally given a brief overview in courses on analysis, algebra, or topology, even though it is sufficiently important, interesting, and simple to merit its own leisurely treatment. This book provides just that: a leisurely exposition for a diversified audience. It is suitable for a broad range of readers, from undergraduate students to professional mathematicians who want to finally find out what transfinite induction is and why it is always replaced by Zorn's Lemma. The text introduces all main subjects of "naive" (nonaxiomatic) set theory: functions, cardinalities, ordered and well-ordered sets, transfinite induction and its applications, ordinals, and operations on ordinals. Included are discussions and proofs of the Cantor-Bernstein Theorem, Cantor's diagonal method, Zorn's Lemma, Zermelo's Theorem, and Hamel bases. With over 150 problems, the book is a complete and accessible introduction to the subject.

集合论的主要概念（基数、序数、超限归纳）对于所有数学家都是最基础的，并非仅限于研究数理逻辑或集合论拓扑的专家。通常分析、代数或拓扑学的课程只会给出基础集合论的一个概貌，然而事实上它足够重要、有趣和简单，值得慢慢地学习品味。 《集合论基础》使得读者能够以悠闲品味的方式学习集合论的内容，它适用于广大范围的各类读者，从本科生直至那些想要最终掌握超限归纳并且理解它为何总被Zorn引理替代的专业数学家。 《集合论基础》介绍了“朴素”（非公理化）集合论的所有主要内容：函数、基数、有序集和良序集、超限归纳及其应用、序数、序数上的运算。《集合论基础》还包括对Cantor-Bernstein定理、Cantor的对角构造、Zorn引理、Zermelo定理和Hamel基的讨论和证明。此外，书中还给出了150多道问题，循序渐进地揭示了集合论基本思想和方法，内容全面完整，具有很好的可读性。

《高等院校计算机专业及专业基础课系列教材·离散数学习题解析》是北京市精品教材《离散数学教程》的配套学习用书，也是北京大学的国家级精品课程“离散数学”的教学参考书。全书由集合论、图论、代数结构、组合数学、数理逻辑等五个部分组成，与《离散数学教程》的教学安排完全一致。 《高等院校计算机专业及专业基础课系列教材·离散数学习题解析》不仅对《离散数学教程》中主要章节的全部习题给出解答，并对《教程》中的重点章节补充了新的习题。这些补充题基本上选自历年教学中的测验考试题，或者研究生入学考题。

《离散数学教程》共分五编。第一编为集合论，其中包括集合的基本概念、二元关系、函数、自然数、基数、序数。第二编为图论，其中包括图的基本概念、图的连通性、欧拉图与哈密顿图、树、平面图、图的着色、图的矩阵表示、覆盖集、独立集、匹配、带权图及其实用。第三编为代数结构，其中包括代数系统的基本概念、几个重要的代数系统：半群、群、环、域、格与布尔代数。第四编为组合灵敏学，其中包括组合存在性、组合计数、级合设计与编码以及组合最优化。第五编为数理逻辑，其中包括命题逻辑、一阶谓词逻辑、Her-brand定理和直觉逻辑。