## The Axiom of Choice

When I read the AC, "that the ∏ of a coll. of non-∅ sets is itself non-∅" I understand its meaning, yet I come short from understanding its cardinal importance in Axiomatic set theory.

I have no exposure "yet" in ZFC but I was hoping if someone could clarify to me why is it that AC is such an important axiom especially that Zermelo used it to formulate the well-ordering theorem. Being also that Set Theory is regarded as the foundation of Mathematics. (Disregarding Godel's work of course)

Thank you
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 AC has a large number of equivalent statements. You touched on well-ordering but there is also "every surjective function has a right inverse", "every non-trivial unital ring has a maximal ideal", "every vector space has a basis", and "two set either have the same cardinality or one is greater than the other."

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 Quote by pwsnafu AC has a large number of equivalent statements. You touched on well-ordering but there is also "every surjective function has a right inverse", "every non-trivial unital ring has a maximal ideal", "every vector space has a basis", and "two set either have the same cardinality or one is greater than the other."
And then there are a myriad of statements which require AC but are not equivalent to it. Some of those statement look pretty innocent. For example:
- A function $f:\mathbb{R}\rightarrow \mathbb{R}$ is continuous at a fixed point x (from the $\varepsilon-\delta$ condition) if and only if for each sequence $x_n\rightarrow x$ holds that $f(x_n)\rightarrow f(x)$. (fun fact: if we change the "at fixed x" by "at every point x", then we don't require AC anymore!)
- For every set X holds that X is either finite or there exists an injection $\mathbb{N}\rightarrow X$
- $\mathbb{N}$ is Lindelof: every open cover of $\mathbb{N}$ has a countable subcover
- $\mathbb{R}$ is not a countable union of countable sets
- Any two bases in a vector space must have the same cardinality
- The Hahn-Banach theorem
- The Ascoli-Arzela theorem
- The existence of the Cech-Stone compactification
- Lebesgue measure is $\sigma$-additive
- Every unbounded subset of $\mathbb{R}$ contains an unbounded sequence

## The Axiom of Choice

 Quote by micromass - For every set X holds that X is either finite or there exists an injection $\mathbb{N}\rightarrow X$ - $\mathbb{N}$ is Lindelof: every open cover of $\mathbb{N}$ has a countable subcover - $\mathbb{R}$ is not a countable union of countable sets - Any two bases in a vector space must have the same cardinality - The Hahn-Banach theorem - The Ascoli-Arzela theorem - The existence of the Cech-Stone compactification - Lebesgue measure is $\sigma$-additive - Every unbounded subset of $\mathbb{R}$ contains an unbounded sequence
These will keep me busy reading for awhile.

 Quote by pwsnafu AC has a large number of equivalent statements. You touched on well-ordering but there is also "every surjective function has a right inverse", "every non-trivial unital ring has a maximal ideal", "every vector space has a basis", and "two set either have the same cardinality or one is greater than the other."
So where does AC play a role here? How did Mathematicians deduce these corollaries from it?
For instance with respect to the right inverse? Where is its role?

Thanks

 Quote by micromass And then there are a myriad of statements which require AC but are not equivalent to it. Some of those statement look pretty innocent. For example: - A function $f:\mathbb{R}\rightarrow \mathbb{R}$ is continuous at a fixed point x (from the $\varepsilon-\delta$ condition) if and only if for each sequence $x_n\rightarrow x$ holds that $f(x_n)\rightarrow f(x)$. (fun fact: if we change the "at fixed x" by "at every point x", then we don't require AC anymore!) - For every set X holds that X is either finite or there exists an injection $\mathbb{N}\rightarrow X$ - $\mathbb{N}$ is Lindelof: every open cover of $\mathbb{N}$ has a countable subcover - $\mathbb{R}$ is not a countable union of countable sets - Any two bases in a vector space must have the same cardinality - The Hahn-Banach theorem - The Ascoli-Arzela theorem - The existence of the Cech-Stone compactification - Lebesgue measure is $\sigma$-additive - Every unbounded subset of $\mathbb{R}$ contains an unbounded sequence
Just finished reading your great blog entries w.r.t. the subject. Very clear and informative read indeed. Thank you.
 Recognitions: Science Advisor Check too, the construction of a nonmeasurable set using AC. Still, I believe non-measurable subsets can be constructed in theories that do not use AC. There is also Tikunov's theorem, Lowenheim-Skolem, which allows you to construct models of the reals of any infinite cardinality (have you heard of the non-standard reals?). For more, see, e.g: http://plato.stanford.edu/entries/ax.../#MatAppAxiCho

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