Before I read this book, I only know ZFC set Theory, and this book is about another set theory called Naive Set Theory, as wikipedia described:
Naïve set theory is any of several theories of sets used in the discussion of the foundations of mathematics. Unlike axiomatic set theories, which are defined using formal logic, naïve set theory is defined informally, in natural language. It describes the aspects of mathematical sets familiar in discrete mathematics (for example Venn diagrams and symbolic reasoning about their Boolean algebra), and suffices for the everyday use of set theory concepts in contemporary mathematics.
While the title states that it is naive, which is usually taken to mean without axioms, the book does introduce all the axioms of ZFC set theory (except the Axiom of Foundation), and gives correct and rigorous definitions for basic objects. Where it differs from a “true” axiomatic set theory book is its character: there are no discussions of axiomatic minutiae, and there is next to nothing about advanced topics like large cardinals. Instead, it tries to be intelligible to someone who has never thought about set theory before.
The whole book(about Naïve set theory) contain eight axioms:
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.
I read this book because set theory is one of the stepping-stone of mathematics, truly understand set theory is necessary to everyone’s mathematics journey:
This book issued 60 years ago, so the symbol system is unfamiliar to me, such as use ∈′ instead of ∉, use - instead of ~ and dozens, quite confused about how symbol system diversity from, and which would be the standard, and a Glossary of set theory is very useful to help you understand all the unfamiliar concepts and different notations
remarkable to me: left side of ∈ is ‘an element’, left side of ⊂ is ‘a set’, and:
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.
