This text for the first or second year undergraduate in mathematics, logic, computer science, or social sciences, introduces the reader to logic, proofs, sets, and number theory. It also serves as an excellent independent study reference and resource for instructors. Adapted from Foundations of Logic and Mathematics: Applications to Science and Cryptography © 2002 Birkhӓuser,this second edition provides a modern introduction to the foundations of logic, mathematics, and computers science, developing the theory that demonstrates construction of all mathematics and theoretical computer science from logic and set theory. The focuses is on foundations, with specific statements of all the associated axioms and rules of logic and set theory, and provides complete details and derivations of formal proofs. Copious references to literature that document historical development is also provided.

Answers are found to many questions that usually remain unanswered: Why is the truth table for logical implication so unintuitive? Why are there no recipes to design proofs? Where do these numerous mathematical rules come from? What issues in logic, mathematics, and computer science still remain unresolved? And the perennial question: In what ways are we going to use this material?Additionally, the selection of topics presented reflects many major accomplishments from the twentieth century and includes applications in game theory and Nash's

equilibrium, Gale and Shapley's match making algorithms, Arrow's Impossibility Theorem in voting, to name a few.

From the reviews of the first edition:

"...All the results are proved in full detail from first principles...remarkably, the arithmetic laws on the rational numbers are proved, step after step, starting from the very definitions!...This is a valuable reference text and a useful companion for anybody wondering how basic mathematical concepts can be rigorously developed within set theory."

—MATHEMATICAL REVIEWS

"Rigorous and modern in its theoretical aspect, attractive as a detective novel in its applied aspects, this paper book deserves the attention of both beginners and advanced students in mathematics, logic and computer sciences as well as in social sciences."

—Zentralblatt MATH

本书适用于数学、逻辑学、计算机科学或社会科学专业的一、二年级本科生，向读者介绍逻辑、证明、集合和数论。它还可以作为优秀的独立学习参考资料和指导教师的资源。改编自《逻辑与数学基础》。该书第二版改编自《逻辑和数学基础：科学和密码学的应用》，© 2002年，Birkhӓuser，对逻辑、数学和计算机科学的基础进行了现代介绍，发展了证明所有数学和理论计算机科学由逻辑和集合论构建的理论。该书的重点是基础，对逻辑和集合论的所有相关公理和规则进行了具体陈述，并提供了形式证明的完整细节和推导。还提供了大量记录历史发展的文献参考。

对许多通常没有答案的问题都找到了答案。为什么逻辑暗示的真值表如此不直观？为什么没有设计证明的秘诀？这些众多的数学规则从何而来？逻辑学、数学和计算机科学中的哪些问题仍未得到解决？还有一个长期存在的问题。此外，所介绍的主题选择反映了二十世纪的许多重大成就，包括博弈论和纳什均衡的应用。

平衡、盖尔和沙普利的匹配算法、阿罗的不可能定理在投票中的应用，仅举几例。

来自第一版的评论。

"......所有的结果都从第一原理中得到了全面详细的证明......值得注意的是，有理数的算术定律从定义开始，一步一步地得到了证明！......对于任何想知道基本数学概念如何在集合理论中得到严格发展的人来说，这是一本有价值的参考书和一个有用的伴侣。"

-数学评论

"在理论方面严谨而现代，在应用方面像侦探小说一样吸引人，这本论文集值得数学、逻辑和计算机科学以及社会科学的初学者和高级学生关注。"

-Zentralblatt MATH