집합론, 그 두 번째 이야기 | 선택공리 ( Axiom of Choice )  By 위대한 멜론빵

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In the previous post, we saw an axiom named axiom of choice.

 

10. Axiom of Choice
S(Sf:SS(A(AS(f(A)A))))For any set S of nonempty sets, there exists a choice function that is defined on S and maps each set of S to an element of the set.

 

Here, it is better to understand S as a family of sets. In other words, it is better to understand S as a set of sets. In addition, S is a set of all elements of elements of S i.e., S=ASA:={xAS,xA}.

 

To put it simply, if not all elements of S are non-empty, we can choose elements one by one from the sets which are the elements of S. For example, if S={{0},{0,1},{0,1,2,3},}, we can select 0 from {0}, 1 from {0,1}, 3 from {0,1,2,3}, and so on.

 

In a way, this seems to be a very obvious result. Of course, if S is a finite set, it would be clearly true. However, in the case of infinite sets, it is not a trivial result.

 

For an infinite set S that doesn't know what kind of set it is, the existence of a choice function for an infinite number of elements becomes quite powerful in future discussions. In addition, there are statements which are equivalent to the axiom of choice used quite often in various discussions, and we'll talk about only three of them which are representative ones.

 

1. Hausdorff Maximal Principle; HMP
In any partially ordered set (P,), (P,) always has a maximal chain.

2. Zorn's Lemma; ZL
If a partially ordered set (P,) has a property that every chain in (P,) has a upper bound, then (P,) has a maximal element.

3. Well Ordering Theorem
Every non-empty set always has an well-ordering.

 

The axiom of choice implies the Hausdorff maximal principle, the Hausdorff maximal principle implies the Zorn's lemma, and the Zorn's lemma implies the well ordering theorem. Finally, the well ordering theorem implies the axiom of choice. We will post the proof of them later.

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