Cantor‘s paradox
In set theory, Cantor’s paradox states that there is no set of all cardinalities. This is derived from the theorem that there is no greatest cardinal number. In informal terms, the paradox is that the collection of all possible “infinite sizes” is not only infinite, but so infinitely large that its own infinite size cannot be any of the infinite sizes in the collection. The difficulty is handled in axiomatic set theory by declaring that this collection is not a set but a proper class; in von Neumann–Bernays–Gödel set theory it follows from this and the axiom of limitation of size that this proper class must be in bijection with the class of all sets. Thus, not only are there infinitely many infinities, but this infinity is larger than any of the infinities it enumerates.
This paradox is named for Georg Cantor, who is often credited with first identifying it in 1899 (or between 1895 and 1897). Like a number of “paradoxes” it is not actually contradictory but merely indicative of a mistaken intuition, in this case about the nature of infinity and the notion of a set. Put another way, it is paradoxical within the confines of naïve set theory and therefore demonstrates that a careless axiomatization of this theory is inconsistent.
Contents
- 1 Statements and proofs
- 2 Discussion and consequences
- 3 Historical notes
1 Statements and proofs
In order to state the paradox it is necessary to understand that the cardinal numbers are totally ordered, so that one can speak about one being greater or less than another. Then Cantor’s paradox is:
Theorem: There is no greatest cardinal number.
This fact is a direct consequence of Cantor’s theorem on the cardinality of the power set of a set.
Proof: Assume the contrary, and let C be the largest cardinal number. Then (in the von Neumann formulation of cardinality) C is a set and therefore has a power set 2C which, by Cantor’s theorem, has cardinality strictly larger than C. Demonstrating a cardinality (namely that of 2C) larger than C, which was assumed to be the greatest cardinal number, falsifies the definition of C. This contradiction establishes that such a cardinal cannot exist.
Another consequence of Cantor’s theorem is that the cardinal numbers constitute a proper class. That is, they cannot all be collected together as elements of a single set. Here is a somewhat more general result.
Theorem: If S is any set then S cannot contain elements of all cardinalities. In fact, there is a strict upper bound on the cardinalities of the elements of S.
Proof: Let S be a set, and let T be the union of the elements of S. Then every element of S is a subset of T, and hence is of cardinality less than or equal to the cardinality of T. Cantor’s theorem then implies that every element of S is of cardinality strictly less than the cardinality of 2T.
2 Discussion and consequences
Since the cardinal numbers are well-ordered by indexing with the ordinal numbers (see Cardinal number, formal definition), this also establishes that there is no greatest ordinal number; conversely, the latter statement implies Cantor’s paradox. By applying this indexing to the Burali-Forti paradox we obtain another proof that the cardinal numbers are a proper class rather than a set, and (at least in ZFC or in von Neumann–Bernays–Gödel set theory) it follows from this that there is a bijection between the class of cardinals and the class of all sets. Since every set is a subset of this latter class, and every cardinality is the cardinality of a set (by definition!) this intuitively means that the “cardinality” of the collection of cardinals is greater than the cardinality of any set: it is more infinite than any true infinity. This is the paradoxical nature of Cantor’s “paradox”.
3 Historical notes
While Cantor is usually credited with first identifying this property of cardinal sets, some mathematicians award this distinction to Bertrand Russell, who defined a similar theorem in 1899 or 1901.
Cantor‘s paradox相关推荐
- wikioi 1083 Cantor表
找规律题 现代数学的著名证明之一是Georg Cantor证明了有理数是可枚举的.他是用下面这一张表来证明这一命题的: 1/1 1/2 1/3 1/4 1/5 - 2/1 2/2 2/3 2/4 - ...
- 康托展开(Cantor expansion)
康托展开是一个全排列到一个自然数的双射.所以可逆. 康托展开:给定一个数n,和一个n位的全排列,求出这个排列是第几位X 逆康托展开:给定一个数n,和这个排列占第几位X, 求出这个排列 这里X(注意第一 ...
- 洛谷——P1014 Cantor表
P1014 Cantor表 题目描述 现代数学的著名证明之一是Georg Cantor证明了有理数是可枚举的.他是用下面这一张表来证明这一命题的: 1/1 1/2 1/3 1/4 1/5 - 2/1 ...
- 算法题——Cantor表
题目介绍 描述 现代数学的著名证明之一是 Georg Cantor 证明了有理数是可枚举的.他是用下面这一张表来证明这一命题的: 1/1, 1/2 , 1/3, 1/4, 1/5, - 2/1, 2/ ...
- 【CodeVS】1083 Cantor表
1083 Cantor表 1999年NOIP全国联赛普及组 时间限制: 1 s 空间限制: 128000 KB 题目等级 : 白银 Silver 题目描述 Description 现代数学的著名证明之 ...
- [NOIP1999] 提高组 洛谷P1014 Cantor表
题目描述 现代数学的著名证明之一是Georg Cantor证明了有理数是可枚举的.他是用下面这一张表来证明这一命题的: 1/1 1/2 1/3 1/4 1/5 - 2/1 2/2 2/3 2/4 - ...
- 洛谷P1014 [NOIP1999 普及组] Cantor 表
现代数学的著名证明之一是 Georg Cantor 证明了有理数是可枚举的.他是用下面这一张表来证明这一命题的: 代码 import java.util.*; public class Main{pu ...
- 【LightOJ - 1104】Birthday Paradox(概率,思维)
题干: Sometimes some mathematical results are hard to believe. One of the common problems is the birth ...
- Cantor定理的一种好表述
今天我在A course on Borel sets 一书中看到了Cantor定理的一种好表述.我很喜欢这种表述.在很多书中,康托定理是这样表述的: 自然数集合的所有子集形成的集合是不可数集. 也有 ...
最新文章
- 2019年,我终于知道86版西游记到底好在哪里了
- 多核之后,CPU 的发展方向是什么?中科院计算所包云岗详细解读
- Pat甲级 1002 A+B for Polynomials
- post json后台处理数据_Python爬虫教程-07-post介绍(百度翻译)(上)
- 怎么隐藏桌面计算机名,Win7隐藏桌面所有图标
- 【分享一套网站源代码】wufowang网源码下载
- 35岁大龄程序员都去哪了?一位网友分享的身边几个真实案例,过于真实!!
- ArrayList 的三种构造方法
- 创业被泼冷水怎么办?
- linux 内核与分析 -- 内存管理(下)
- Alexa工具条正确安装方法 Alexa工具条正确使用方法 Alexa工具条正确应用
- 程序员该如何写好简历
- 数据可视化Matplotlib库
- 天眼查 Authorized和企查查 sign破解
- AI中数据标注(Labeling)的介绍
- 求华摄氏度100‘F对应下的摄氏温度
- C#设计模式学习笔记:(4)建造者模式
- Java关键字-static
- 企业注销要谨慎,这些注意事项你要知道
- springboot 到底有什么魅力?