Book Review: Naive Set Theory

作者：Paul R. Halmos
出版社： D. Van Nostrand Company, Inc.
发行时间：1960
来源：下载的 epub 版本
Goodreads：4.21 (351 Ratings)
豆瓣：8.8（50人评价）

Every mathematician agrees that every mathematician must know some set theory; the disagreement begins in trying to decide how much is some. This book contains my answer to that question. The purpose of the book is to tell the beginning student of advanced mathematics the basic set-theoretic facts of life, and to do so with the minimum of philosophical discourse and logical formalism. The point of view throughout is that of a prospective mathematician anxious to study groups, or integrals, or manifolds. From this point of view the concepts and methods of this book are merely some of the standard mathematical tools; the expert specialist will find nothing new here.

Observe that belonging (∈) and inclusion (⊂) are conceptually very different things indeed. One important difference has already manifested itself above: inclusion is always reflexive, whereas it is not at all clear that belonging is ever reflexive. That is: A ⊂ A is always true; is A ∈ A ever true? It is certainly not true of any reasonable set that anyone has ever seen. Observe, along the same lines, that inclusion is transitive, whereas belonging is not. Everyday examples, involving, for instance, super-organizations whose members are organizations, will readily occur to the interested reader.

Axiom of extension. Two sets are equal if and only if they have the same elements.
Axiom of specification. To every set A and to every condition S(x) there corresponds a set B whose elements are exactly those elements x of A for which S(x) holds.
Axiom of pairing. For any two sets there exists a set that they both belong to.
Axiom of unions. For every collection of sets there exists a set that contains all the elements that belong to at least one set of the given collection.
Axiom of powers. For each set there exists a collection of sets that contains among its elements all the subsets of the given set.
Axiom of infinity. There exists a set containing 0 and containing the successor of each of its elements.
Axiom of choice. The Cartesian product of a non-empty family of nonempty sets is non-empty.
Axiom of substitution. If S(a, b) is a sentence such that for each a in a set A the set {b: S(a, b)} can be formed, then there exists a function F with domain A such that F(a) = {b: S(a, b)}for each a in A.
Counting theorem. Each well ordered set is similar to a unique ordinal number.