[Naive Set Theory] Section 1 The Axiom of Extension

Naive Set Theory

Section 1 The Axiom of Extension


前置要点 Highlights

P. 1 Para. 1: The word "class" is also used in this context, but there is a slight danger in doing so. The reason is that in some approaches to set theory "class" has a special technical meaning.

  • 慎用类一词叙述

P. 1 Para. 2: One thing that the development will not include is a definition of sets. The situation is analogous to the familiar axiomatic approach to elementary geometry.
P. 1 Para. 2: The semi-axiomatic point of view adopted here assumes that the reader has the ordinary, human, intuitive (and frequently erroneous) understanding of what sets are.

  • 集合的定义实际并不明确.

P. 1 Para. 3: Sets, as they are usually conceived, have elements or members.

  • 集合中有元素.

P. 1 Para. 3: It is important to know that a set itself may also be an element of some other set.

  • 一个集合可以是另一个集合的元素.
  • 隐含了有关第三次数学危机的内容,也是朴素集合论的缺陷.

阐明 Exposition

  • 关于字母的一些约定俗成惯例

  • 属于 belonging
    $x$属于$A$$x$$A$的一个元素),则写作$x\in A$.

  • 相等 equality

    • 集合之间可能的比属于更基本的关系就是相等.
    • $A$$B$相等仍然是以熟悉的记号表示为$A=B$;
    • 同样地,$A$$B$不相等表示为$A\neq B$.
  • 属于相等的联系
    • 外延公理. 两个集合相等当且仅当它们拥有相同的元素.
    • 稍微有逼格一点,集合由其外延确定.
  • 子集包含 subset & inclusion

    • $A$$B$为集合,且$A$的每个元素同样也是$B$的元素之一,那么$A$$B$的一个子集,即$B$包含$A$,表示为$A\subset B$或者$B \supset A$.
    • $A$$B$为集合,且$A\subset B, A \neq B$,那么$A$$B$真子集.
  • 一些性质

    1. (自反性 reflexive)$A\subset A$.
    2. (传递性 transitive)$A, B, C$都是集合且$A\subset B, B\subset C$,那么$A\subset C$.
    3. (对称性 symmetric)$A=B \Leftrightarrow B=A$.
  • 外延公理的重新叙述

    • 外延公理. 若$A$$B$为集合,$A = B \Leftrightarrow A\subset B, B\subset A$.
    • 要证明集合$A$$B$相等,也一般从$A\subset B$$B\subset A$两个部分入手.
  • 辨析
    • 属于($\in$)和包含($\subset$)其实是截然不同的东西,但需要注意包含具有自反性,即$A\subset A$恒成立;反观属于$A\in A$,这是不可能的.
    • 相关内容可延伸至第三次数学危机.