# The Axiom of Choice

> 【EN】The principle of set theory known as the Axiom of Choice has been hailed as “probably the most interesting and, in spite of its late appearance, the most discussed axiom of mathematics, second only to Euclid’s axiom of parallels which was introduced more than two thousand years ago” (Fraenkel, Bar-Hillel & Levy 1973, §II.4). The fulsomeness of this description might lead those unfamiliar with the axiom to expect it to be as startling as, say, the Principle of the Constancy of the Velocity of Light or the Heisenberg Uncertainty Principle. But in fact the Axiom of Choice as it is usually stated appears humdrum, even self-evident. For it amounts to nothing more than the claim that, given any collection of mutually disjoint nonempty sets, it is possible to assemble a new set—a transversal or choice set —containing exactly one element from each member of the given collection. … 【中】词条介绍选择公理的起源与年代线索，说明它相对于其他集合论公理的独立性与一致性结果，考察与之等价的极大原理如佐恩引理，并展示它在数学各分支中的应用。读者会看到，一条看似自明的断言如何在证明中承担远超直觉的重任。

- ID: m13155
- Category: decide
- Domain: 决策科学

## Definition

选择公理是集合论中的一条原则：给定一族非空且两两不交的集合，存在一个集合恰好从每个集合中各取一个元素。它在数学中出现较晚，却是除欧几里得平行公理之外被讨论最多的公理，而其表述看上去平淡甚至自明。 脚手架作用：- 辨析存在：区分断言某物存在与给出找到它的方法。 - 检验隐含前提：发现证明中悄悄做了无穷次无规则选取。 - 权衡后果：提醒接受它会带来分球怪论等反直觉结论。

## How it works

争议在于选取的两种方式：若能用一条规则指明取哪个元素，就不需要这条公理；但若有无穷多个集合且没有任何规则可循，逐个选取就无法完成。选择公理直接断言这样一个集合存在却不给出构造方法，因此它保证存在性却不提供计算途径，这正是它既自明又可疑的原因。

## Practice

1) 检查你要做的选取是有限次还是无穷次；2) 若能给出明确的选取规则，就用规则完成，无需动用公理；3) 若涉及无穷多集合且无规则可循，明确承认使用了选择公理；4) 留意结论是否只保证存在而无法构造，避免把存在性当成可计算性。

## Use

- 辨析存在：区分断言某物存在与给出找到它的方法。 - 检验隐含前提：发现证明中悄悄做了无穷次无规则选取。 - 权衡后果：提醒接受它会带来分球怪论等反直觉结论。

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