# 1. 概述

## 1.1 集合论地位

Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice[1].

## 1.2 现代集合论历史[1]

（1）朴素集合论

（2）罗素悖论——第三次数学危机

（3）公理化集合论

In 1908, two ways of avoiding the paradox were proposed, Russell’s type theory and the Zermelo set theory, the first constructed axiomatic set theory. Zermelo’s axioms went well beyond Frege‘s axioms of extensionality and unlimited set abstraction, and evolved into the now-canonical Zermelo–Fraenkel set theory (ZFC).

The essential difference between Russell’s and Zermelo’s solution to the paradox is that Zermelo altered the axioms of set theory while preserving the logical language in which they are expressed (the language of ZFC, with the help of Skolem, turned out to be first-order logic) while Russell altered the logical language itself.

# 2. 一些基本概念

（1）ZFC

In mathematics, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is one of several axiomatic systems that were proposed in the early twentieth century to formulate a theory of sets free of paradoxes such as Russell’s paradox.

Zermelo–Fraenkel set theory with the historically controversial axiom of choice (选择公理) included is commonly abbreviated ZFC, where C stands for choice (Ciesielski 1997). Today ZFC is the standard form of axiomatic set theory (公理集合论) and as such is the most common foundation of mathematics.

（2）集合（set）

（3）关系（relation）

（4）函数（function）

（5）等价（equivalence）

（6）选择公理（Axiom of Choice）

The axiom of choice, or AC, is an axiom of set theory equivalent to the statement that the Cartesian product (笛卡尔积) of a collection of non-empty sets is non-empty.

It states that for every indexed family of nonempty sets there exists an indexed family of elements such that for every .

To give an informal example, for any (even infinite) collection of pairs of shoes, one can pick out the left shoe from each pair to obtain an appropriate selection, but for an infinite collection of pairs of socks (assumed to have no distinguishing features), such a selection can be obtained only by invoking the axiom of choice.

# 3. 学习资料

Ernst Zermelo, Abraham Fraenkel. Zermelo–Fraenkel set theory with the axiom of choice (ZFC). Charles E. Merrill Publishing Company (1968)

Today ZFC is the standard form of axiomatic set theory and as such is the most common foundation of mathematics[2].

Nicolas Bourbaki. Elements of Mathematics: Theory of Sets.1970(English translation).

James Munkres. Topology(2nd Edition). Pearson.2010.

[1]维基百科词条：集合论,
[2]维基百科词条：Zermelo–Fraenkel set theory

### 2 thoughts on “集合论学习笔记”

• 2015年11月05日 星期四 at 01:06上午