Explain Axiom of Choice in Simple Terms

[Vadermort]

Can someone pretty please explain to me the axiom of choice as unrigorously and casually as possible ;P.
And then you could include a rigorous explanation if you wish, but I mean I could just go to wikipedia for that..which I already did... in fact I think I'll include the wiki explanation and parts I have trouble with...
FROM WIKIPEDIA:
First statement of the axiom (the very beyond me explanation):
for every family (Si)i ∈ I of nonempty sets there exists a family (xi)i ∈ I of elements with xi ∈ Si for every i ∈ I.
1. ∈ <---what is this?
2. What about the rest of it? (really just don't have a firm grasp of the notation here, the only domains of math I have experience with are middle school algebra and calculus)

Second statement (the slightly less beyond me explanation):
A choice function is a function f, defined on a collection X of nonempty sets, such that for every set s in X, f(s) is an element of s. With this concept, the axiom can be stated:
For any set X of nonempty sets, there exists a choice function f defined on X.

1.) Is a collection of sets the same thing as a set of sets? or are sets defined by certain qualities e.g the natural numbers constitute a set...
2.) my attempt at restating it: for every set s in X, f(X) will yield an f(s) ...and why is this important?

 

[jhae2.718]

By our very own Micromass:
https://www.physicsforums.com/blog.php?b=2951

 

[micromass]

Ah, the axiom of choice. My favorite subject

May I ask you why you're interested in the axiom of choice? It's just that it is very hard to grasp for somebody who just knows middle school algebra. I just want to see where you're comming from.

Can someone pretty please explain to me the axiom of choice as unrigorously and casually as possible ;P.
And then you could include a rigorous explanation if you wish, but I mean I could just go to wikipedia for that..which I already did... in fact I think I'll include the wiki explanation and parts I have trouble with...
FROM WIKIPEDIA:
First statement of the axiom (the very beyond me explanation):
for every family (Si)i ∈ I of nonempty sets there exists a family (xi)i ∈ I of elements with xi ∈ Si for every i ∈ I.
1. ∈ <---what is this?

This means "is an element from". For example $$5\in \{1,2,3,5\}$$ means that 5 is contained in the set with {1,2,3,5}. On the other hand $$5\notin \{1,2,3\}$$ means that 5 is not contained in {1,2,3}.

2. What about the rest of it? (really just don't have a firm grasp of the notation here, the only domains of math I have experience with are middle school algebra and calculus)

I suggest you read a good book on set theory. It's not very hard, it are just weird notations. A more detailed explanation without notations can be found on my blog...

Second statement (the slightly less beyond me explanation):
A choice function is a function f, defined on a collection X of nonempty sets, such that for every set s in X, f(s) is an element of s. With this concept, the axiom can be stated:
For any set X of nonempty sets, there exists a choice function f defined on X.

1.) Is a collection of sets the same thing as a set of sets? or are sets defined by certain qualities e.g the natural numbers constitute a set...

Yes, a collection of sets is exactly the same thing as a set of sets.

2.) my attempt at restating it: for every set s in X, f(X) will yield an f(s) ...and why is this important?

There is no such thing as f(X). f is defined only on elements on X. Thus you can have f(s) for every set s in X. But you cannot have f(X), since X is not an element of X.

 

[Vadermort]

"Suppose that there are an infinite number of prisoners who are all to face the firing squad. The jail captain states the conditions of the game: "On the day you will face the firing squad, I will put a black or a white hat on every one of you. You can look around, but cannot speak/gesture to each other. You will not be told the color of your own hat. When I give the signal, you will all simultaneously shout out the color you believe your hat to be. If an infinite number of you are wrong, you will all be shot dead. If only a finite number of you are wrong, you all live."
The night before the shooting, the prisoners gather 'round, trying to concoct a strategy to live. As they cannot communicate with each other on the day they receive their hats, they need to formulate a plan first. Does such a strategy exist? If so, what is it?
Note: the prisoners can look around them in line and have infinite vision (to see everyone else), as well as infinite memory."
One of the hints given to me was that the axiom of choice was involved somehow. Thanks for the explanation!

 

[micromass]

This is a so-called infinite hat problem ans such things are closely related to the axiom of choice. See http://cornellmath.wordpress.com/2007/09/13/the-axiom-of-choice-is-wrong/ for a nice explanation!

 

[Vadermort]

P.s. is linear algebra a good preface to set theory or is my current level of mathematical education lacking in basic logical rigor? :P

 

[micromass]

P.s. is linear algebra a good preface to set theory or is my current level of mathematical education lacking in basic logical rigor? :P

Set theory is a ridiculous easy theory to learn. In a way, you need no prerequisites for it. However, set theory is commonly taught in graduate courses, for quite a few reasons:

1) You need to comfortable with some notations and terminology. For example, not knowing what $$\in$$ means, indicates that you're not yet familiar with the basic notations, thus you should familiarize a bit.

2) Most important: if you take set theory now, you might understand it all if you work hard. However, you won't have the slightest clue for what it's all good for. Why would something like Zorn's lemma ever be useful? Why do we even care about sets and countability and stuff?? Things like this indicate that you need some more experience.

I would suggest that you take a few math (3 or 4, more is better) courses before taking set theory. These can be ANY courses, but the courses need to be proof-based, and need to be quite rigourous.
Courses like abstract algebra, topology and real analysis are good courses to build experience. So take at least one of these... And again, the more experience you have, the better.

A good book that you could read right now is "Essentials of mathematics" by Hale. You will see a glimpse of set theory in there...

 

[Vadermort]

thanks^3

 

[quantum123]

How to prove Zorn's lemma from Axiom of Choice?

 

[disregardthat]

This is a so-called infinite hat problem ans such things are closely related to the axiom of choice. See http://cornellmath.wordpress.com/2007/09/13/the-axiom-of-choice-is-wrong/ for a nice explanation!

I am using the opportunity to bump this interesting riddle

https://www.physicsforums.com/showthread.php?t=476964

Don't read the link if you don't want the answer. It is the infinite part that is relevant to the axiom of choice.

 

[micromass]

How to prove Zorn's lemma from Axiom of Choice?

I've answered this twice already... What don't you understand about this proof: https://www.physicsforums.com/showthread.php?t=488969&highlight=Zorn

 

[quantum123]

But I cannot open the file. Help!