集合论学习笔记

本文系为学习集合论的一些笔记。

1. 概述

1.1 集合论地位

博文《MIT牛人解说数学体系》(已转至本站,这里)说集合论是现代数学的共同基础,在此基础上建立数学两大家族:分析、代数。

集合论是研究集合(由一堆抽象对象构成的整体)的数学理论,包含集合和元素(或称为成员)、关系等最基本数学概念。大多数现代数学的公式化都是在集合论的语言下谈论各种数学对象,所以集合论常被视为数学基础之一,以未定义的“集合”与“集合成员”等术语形式化建构数学对象。(集合论、命题逻辑与谓词逻辑共同构成了数学的公理化基础)[1]

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

现代集合论的研究开始于1870年代由俄国数学家康托尔及德国数学家理察·戴德金的朴素集合论(naive set theory)开始。朴素集合论的开始是1874年康托尔的一篇论文《On a Characteristic Property of All Real Algebraic Numbers》。在朴素集合论中,集合是当做一堆对象构成的整体之类的自证概念(any definable collection is a set),没有有关集合的形式化定义

(2)罗素悖论——第三次数学危机

在1900年左右许多数学家发现朴素集合论会产生一些矛盾的情形,称为二律背反或是悖论伯特兰·罗素(1901年发现)和恩斯特·策梅洛(1900年发现,但没发表)均发现了最简单的悖论。罗素悖论(Russell’s paradox)也叫理发师悖论,即:

在村有一位手艺高超的理发师,他只给村上一切不给自己刮脸的人刮脸,那麽,他给不给自己刮脸呢?如果他不给自己刮脸,他是个不给自己刮脸的人,他应当给自己刮脸;如果他给自己刮脸,由於他只给不给自己刮脸的人刮脸,他就不应当给自己刮脸了。他应该如何呢?

考虑“由所有不包含集合自身的集合所构成的集合”,记之为S。不管假设S是或不是S自身的元素,按照S的定义都会导致矛盾。罗素悖论也造成了第三次数学危机。

(3)公理化集合论

在发现朴素集合论会产生一些悖论后,二十世纪初期提出了许多公理化集合论,其中最著名的是包括选择公理的策梅洛-弗兰克尔集合论,简称ZFC。公理化集合论不直接定义集合和集合成员(每个元素被视为一个集合),而是先规范可以描述其性质的一些公理。摘抄维基百科词条Russell’s paradox如下:

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

策梅洛-弗兰克尔集合论(Zermelo-Fraenkel Set Theory),含选择公理时常简写为ZFC,是数学基础中最常用形式公理化集合论,不含选择公理的则简写为ZF。摘抄维基百科词条Zermelo–Fraenkel set theory部分如下:

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)

选择公理是指任意的一群非空集合,一定可以从每个集合中各拿出一个元素。摘抄维基百科词条Axiom of Choice部分如下:

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 (S_{i})_{i\in I} of nonempty sets there exists an indexed family (x_{i})_{i\in I} of elements such that x_{i}\in S_{i} for every i\in I.

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.

 

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

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2 thoughts on “集合论学习笔记

  • 2015年11月04日 星期三 at 05:06下午
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    请问您搞的网络编码研究都需要哪些数学基础哪?能否指点下

    Reply