# 朴素集合论

## 集合、成员及相等

### 成员

x是集合A的成员，也可以说x属于A，可以用x ∈ A表示，∈符号衍生自希腊字母小写的ε，是朱塞佩·皮亚诺在1889年引入，应该是因为是ἐστί（意思是"是"）的第一个字母。也常在x ∉ A的式子中用到符号 ∉，意思是x不属于A。

## 特点

### 悖论

• 若此集合不是集合本身的成员，此集合符合“不包括自身的集合”的定义，应该要是此集合的成员之一，矛盾。
• 若此集合是集合本身的成员，此集合不符合“不包括自身的集合”的定义，不应该在此集合中，矛盾。

## 脚注

1. ^ Concerning the origin of the term naive set theory, Jeff Miller says, "Naïve set theory (contrasting with axiomatic set theory) was used occasionally in the 1940s and became an established term in the 1950s. It appears in Hermann Weyl's review of P. A. Schilpp (ed) The Philosophy of Bertrand Russell in the American Mathematical Monthly, 53., No. 4. (1946), p. 210 and Laszlo Kalmar's review of The Paradox of Kleene and Rosser in Journal of Symbolic Logic, 11, No. 4. (1946), p. 136. (JSTOR)." [1] The term was later popularized by Paul Halmos' book, Naive Set Theory (1960).
2. ^ Cantor 1874
3. ^ Frege 1893 In Volume 2, Jena 1903. pp. 253-261 Frege discusses the antionomy in the afterword.
4. ^ Peano 1889 Axiom 52. chap. IV produces antinomies.