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Just as the title stated, what is the main point of axiom of choice? I do not quite understand what is written in the axiom. The axiom that I know is:

Given any collection of non-empty sets, there exists a choice function such that $$f:I \rightarrow \bigcup_i{S_i}\quad f(i)\in S_i \quad\text{for all }\; i\in I.$$

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The point of the Axiom of Choice is that oftentimes a mathematician finds him- or herself at a point where infinitely many specific objects must be gathered up all at once in order to continue a proof/construction.

In the statement as given, we have a family of nonempty sets $\{ A_i : i \in I \}$, and we want to pick a unique representative from each $A_i$, this is our function $f : I \to \bigcup_{i \in I} A_i$ ($f$ "picks" $f(i)$ to be the unique representative from $A_i$). The Axiom of Choice says that this is unproblematic, and we can always do this.

Of course, there are certain specific instances where one can do this without appealing to the Axiom of Choice:

  • if you only have to make finitely many choices; or
  • if these choices can be made in a uniform manner (e.g., if I have infinitely many nonempty sets of natural numbers, I can choose the least one from each set).

More often there is no way to uniformly pick these representatives, and without appealing to some extra-logical hypothesis we cannot make the choices as required.

(There are many statements known to be equivalent to the Axiom of Choice. The most common one you see in mathematics outside of logic/set theory is Zorn's Lemma.)

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The axiom of choice states that when given a family of nonempty sets, we can choose an element from each set. Hence the name, the axiom of choice.

It is useful because it allows us to control the behavior of infinite objects. It ensures that we can prove the existence of things which otherwise may require an infinitely long statement, for example a Hamel basis for every vector space.

When writing a proof about an infinite collection of objects we will sometimes want to make some arbitrary choice amongst them and just "run with it". It might be possible to make that choice less arbitrary, for example if we talk about finite sets of real numbers you can always take the minimal element; or if you talk about continuous functions from a compact interval to $\mathbb R$, you can talk about the smallest point which attains the maximal value.

When you have a uniform way of making a choice, that is one "algorithm" which ensures that a choice is made, then you really have no need for the axiom of choice. The axiom of choice supplies this method when you cannot prove it exists otherwise.


Further reading:

  1. Simple and intuitive example for Zorns Lemma
  2. Making a choice function if $A_i$ are well-ordered for each $i$
  3. Is there any motivation for Zorn's Lemma?
  4. Where is the Axiom of choice used?
  5. Does proving (second countable) $\Rightarrow$ (Lindelöf) require the axiom of choice?
  6. Axiom of Choice Examples
  7. A concrete example of a choice function
  8. Ultra Filter and Axiom of Choice
  9. Every Hilbert space has an orthonomal basis - using Zorn's Lemma
  10. Is Banach-Alaoglu equivalent to AC?
  11. Axiom of choice and compactness.
  12. Is the axiom of choice needed to show that $a^2=a$?
  13. Advantage of accepting non-measurable sets
  14. Algebraic closure for $\mathbb{Q}$ or $\mathbb{F}_p$ without Choice?
  15. A question concerning on the axiom of choice and Cauchy functional equation
  16. Advantage of accepting the axiom of choice

Some of these threads discuss implications of the axiom of choice, or its common equivalents the well-ordering principle and Zorn's lemma. Some of them discuss where these principles are used and what may happen when they fail.

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It says that given a collection of non-empty sets, you can choose an element from each set.

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It's just an axiom that lets you choose one element out of each set in a collection. The choice function maps each index to some element in the corresponding set. The idea is that you're going from set to set in your collection, and picking one thing out of each one.

If you have ten bins of stuff, it's not unreasonable to suppose you can pick one thing out of each bin. That's all the axiom of choice says --- there's not much beyond its name.

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The axiom of choice is not needed to pick one thing out of each bin if there are only ten bins, all of which are non-empty. That's actually a fairly straightforward consequence of the definition of non-emptiness. The problem is when you have infinitely many bins... –  Zhen Lin Jan 8 '13 at 21:01
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