Axiom of choice?

Hmm, I can see why your explanation might not be convincing. One can choose a marble from a bag without ever learning anything about its colouring. And one can choose a real number quite easily -- 36 happens to be a personal favourite.

The tricky part about the axiom of choice is that it doesn't in any way restrict the (cardinal) number of sets from which one can choose elements, which is counterintuitive to how one tends to make calculations, or argue proofs, in finite steps. And it doesn't require us to construct our choice function -- we don't have to know what we're doing.

So, the argument I'd go with would be in terms of trying to pick a marble from each one of an infinite number of bags -- we'd never finish. Or perhaps I might try it in terms of picking an invisible marble from an intangible bag. Or a combination of the two.

u can understand the basic concept of axiom from the following explanation.......

The tricky part about the axiom of choice is that it doesn't in any way restrict the (cardinal) number of sets from which one can choose elements, which is counterintuitive to how one tends to make calculations, or argue proofs, in finite steps. And it doesn't require us to construct our choice function -- we don't have to know what we're doing.

So, the argument I'd go with would be in terms of trying to pick a marble from each one of an infinite number of bags -- we'd never finish. Or perhaps I might try it in terms of picking an invisible marble from an intangible bag. Or a combination of the two

Hope this Helps u!!!!!!!!!!

Before I try to give another (better(?)) example, let me try to get clear on what the Axiom of Choice states:

Let C be a collection of nonempty sets. Then we can choose a member from each set in that collection. In other words, there exists a function f defined on C with the property that, for each set S in the collection, f(S) is a member of S.

That's about the simplest definition I know. Why is it (or why was it) controversial? What if I cannot clearly specify what "f" actually is? Here is what Bertrand Russel had to say about it:

"To choose one sock from each of infinitely many pairs of socks requires the Axiom of Choice, but for shoes the Axiom is not needed."

The idea is that the two socks in a pair are identical in appearance, and so we must make an *arbitrary* choice if we wish to choose one of them. For shoes, we can use an explicit algorithm -- e.g., "always choose the left shoe." Why does Russell's statement mention infinitely many pairs? Well, if we only have finitely many pairs of socks, then the Axiom of Choice is not needed -- we simply choose one member of each pair using the definition of "nonempty," and we can repeat an operation finitely many times.

The Axiom of Choice, then, becomes the "intuitive" way we think about choosing: when we are faced with trying to select objects from sets, and we cannot actually write down *how* we are going to choose, we just choose anyway.

So, here's my example (it is a little long):

I have a bag with an infinite number of stuffed bunnies, some with two white ears, some with two black ears and some with one ear of each color. Can I write down on a piece of paper the exact rule to select just one bunny? Yes, easy: pick the first bunny with a black left ear. Done!

Now I have another bag same as the first, but with teddy bears. Can I write down a rule that lets me pick a teddy bear *and* a bunny? Sure. It's the same rule but slightly expanded: From the first bag pick the first bunny with a black left ear, and from the second bag pick the first bear with a black left ear. Done!

Now I add a bag of lizards. Oops, lizard don't have external ears. Hmmm. I modify the rule to pick a bunny, a bear and the first lizard with a black tail. My rule is getting longer.

Now, I add a bag of pairs of socks. Ugh. What do I need a rule for! Just open the bag and pick one, darn it!

*THAT"S* the Axiom of Choice!

HTH

Charles

Yes, if you say 1, you could be talking about something infinetly close to 1, like 0.99999999999999999999999999999999999999999999999

or

1.00000000000000000000000000000000000000000000001

Please give me best answer thanks!