preliminary remark: the axiom of choice ( Auswahlaxiom in Germany) can be formulated this way:
For all sets M there is a selection function, that assigns for all elements of the power set P(M) exept ∅ an element of the corresponding subset of M.
It is assumed to be true in many areas of mathematics. Besides its "job" of giving elements of infinite sets, it has equivalent formulations to give "upper bounds" (Lemma of Zorn) and others. It is crucial in functional analysis in many ways.
In the following dialogue I tried to clarify a bit to myself, what this axiom means and why I do not entirely trust it.
Beware: It is an axiom, not a proven theorem. All that is proven about it is the following: It can be included in certain axiom systems without generating a contradiction.
The mathematical god of set theory
There is a little god in mathematics, who is known by many names. The Lemma of Zorn is one, but I choose the name axiom of choice.
Both are equivalent, which is "the same" mathematicaly.
This little god does many things in many mathematical areas.
I choose its job in set theory.
Every time a mathematician says:
"I choose an element of the set M to do this and that with",
it takes an element and gives the element to her - or, in some cases, him or it. I mean, how else can the mathematician get it?
So a person enters the temple of choice and asks:
person: "O axiom of choice, give me an element of the set M."
axiom : "I, the mighty axiom of choice, have a question first.
Did you ensure the set M is not empty? Otherwise I will not give you an element of the set M. I will never do this.
So I ask you: Are you sure the set M is not empty? By the way, which set M are we talking about?"
person: "I want a dialog between the axiom of choice and a mathematician. But I am too lazy to write it. Give me a mathematician to write a dialog with you."
axiom: "Any special kind of mathematician?"
person: "How about Douglas Hofstadter? He is very good at writing dialogs."
axiom: " The person, who entered my temple, wishes an element of a subset of mathematicians. Is this set not empty?"
person: "Of course not. It has one element."
axiom: "The petitioner wishes an element of a subset with one element. I give you Douglas Hofstadter."
person: "Where is he?"
axiom: "I give elements in a more mathematical sense. To find it is an exercise for students.
Normally I would not do this, but since you are talking to your god, I give you a little hint: Look at his e-mail or his telefon number."
person: "I am too lazy, Besides, I do not know if he still lives."
axiom: "In that case, take a shovel and a good book of necromancy.
By the way - is this the old trick to call on a god and then explain to it that it does not exist? Won´t work. I am even a proven theorem for finite sets!"
person: "That is the trouble, you are even trivial - in a finite set. There you are a theorem, not an axiom."
axiom: " After I was formulated in 1904 by Ernst Zermelo, I was even gödelized! I work for infinite sets too. You will never be able to disprove me!"
person: "Yeah, that is the trouble. Axioms can not be proven or disproven. They have to be believed. I don´t believe in you."
axiom: "I knew it. Always the same trick. Now - you wish for a mathematican for your dialog? Will any mathematican do? I can bring you elements of subsets too."
person: "Then I wish for a constructive mathematican!"
axiom: "Why a constructive mathematican?"
person: "I think a constructive mathematician is a mathematician, who does not believe in you."
axiom: "Thought so. Oh person in my temple, I ask you: Is this subset of mathematicans not empty?"
person: "Of course not. All the old greek mathematicians didn´t believe in you!"
axiom: "Wait a minute. I was formulated 1904. How could they not believe in me? I did not even exist then."
person: "You were not formulated then because they did not believe in you."
axiom: "Why do you think that?"
person: "Because they hadn´t the decimal system or even the old babylonian sexagesimal system, because they had these questions about achilles and the turtle, of the doubling of the altar of apollo, about π, and .."
axiom: "Okay, okay. You want an Element of this subset of all mathematicans : Old greek mathematicians."
person: "Yes, give me one. "
(I hope it says Eukild, because then Euklid could explain this axiom thing and ... )
axiom: "Archimedes".
(Hmm. What can I do with that one - Construction or - which mathematician did it pick again?)
person: "Sorry whom did you coose again?"
axiom: "Phytagoras."
person: "What? I think you said -"
axiom: " My selection function was on the loo. Besides all mathematicans say it does not matter which element of the subset I give them."
person : " What? I begin to think I am out of my depth here, trying out what you do.
Let´s take a look at how you are formulated instead. Socratean method of course. Now I read something about subsets here. So: What is a subset?
axiom: "An Element of the power set of M."
person: "Can you explain what a power set is?"
axiom: "Not my job, but okay. Have you a set M?"
person: "We could use glas beads. I have lots of them. I could even build an euclidian geometry out of four beads and six strings!"
axiom: "Okay, let us take four boxes. Take a red and a green bead. One box is empty. Now take a red bead and - why are you looking in the empty box?"
person: " I want to see if it is still empty."
axiom: "Okay, okay no boxes. You have lots of transparent bags here. Take four of them. One is empty. Now put a red bead in a bag."
person: "Okay."
axiom: "Put a green one in the next bag."
person: "Okay."
axiom: "Now put a red bead and a green bead into the next bag".
person: "Okay."
axiom: "What do you have?"
person: " Three empty bags and a bag with a red and a green bead in it."
axiom: " What? You should not have taken the red and green beads out of their bags!"
person: "How else could I have put them into the last bag?"
axiom: "You have lots and lots of red and green beads here! Take another red and another green one!"
person: "That are not the same beads. Okay, okay, I will do it."
axiom: "Now you have the potency quantity of the set of a red and a green bead."
person: "The power set is a set too, yes?"
axiom: "Yes"
person: "I could build the power set of that set too! I only need more of these litttle bags, bigger bags and more glas beads! And the power set of that and so on and on!"
axiom: "Yes."
person: "I wonder how many red and green beads I have. I should buying some at the bead shop.
I wonder how many beads they have?
How many grains of sand, I mean red and green beads are in the universe? But back to this selection function. What is that?"
axiom: "A selection function is a function, that chooses for any subset of a set M - exept the empty one - an element of that subset.
I, the mighty axiom of choice assure you, that there is such a function for every set M."
person: "I begin to see, why you are a trivial theorem for a finite set M. But back to the subset of greek mathematicians. You said the function was on the loo? And suddenly there was another?"
axiom: "I never said I had only one function. Mostly I have lots of them. Choose one."
person: "Give me one."
axiom: "The petioner whishes a selection function for the power set P(mathematicians). This is a function from P(P(mathematicians)) to P(mathematicians).
It seems any one will do. Here you have one."
person: "You are sure there is one? Could the set of this selection functions not be empty?"
axiom: "Yes, that is what I am: I assure you there is always a selection function for every set M. This is the one case where you never have to worry about an empty set. I will always guarantee that. Have you any special wishes for subsets?"
person: "Just give me one! And this time a unique one. "
axiom: " Then you must ask me about an element of a subset with one element again. I will spare you that discussion and give you a non-special one of the set.
The petioner still whishes a unspecified selection function from the potency quantityP(P(mathematicians)) in the potency quantitiy P(mathematicians) . Here."
person: "I begin to see how that goes on. Just give me the nearest constructive mathematician. That will do."
axiom: "That is not my job!"
person: "I think it is, when you are in the disguise of the Lemma of Zorn."
axiom: "Okay. Nearest constructive mathematician - what do you think a constructive mathematician does, execpt not believeing in me?"
person: "Building axiom systems comes to mind."
axiom: "Have you, coincidencly, build an axiom system? I remember this little euclidian plane?"
person: "That was an example. But yes, I have build an axiom system around it. And I think it works even for infinite euclidean geometries without needing you!"
axiom: "I will now give you the maximal nearest constructive mathematician you could use."
person: "But I am too lazy to write this dialogue!"
preliminary remark: the axiom of choice ( Auswahlaxiom in Germany) can be formulated this way:
For all sets M there is a selection function, that assigns for all elements of the power set P(M) exept ∅ an element of the corresponding subset of M.
It is assumed to be true in many areas of mathematics. Besides its "job" of giving elements of infinite sets, it has equivalent formulations to give "upper bounds" (Lemma of Zorn) and others. It is crucial in functional analysis in many ways.
In the following dialogue I tried to clarify a bit to myself, what this axiom means and why I do not entirely trust it.
Beware: It is an axiom, not a proven theorem. All that is proven about it is the following: It can be included in certain axiom systems without generating a contradiction.
The mathematical god of set theory
There is a little god in mathematics, who is known by many names. The Lemma of Zorn is one, but I choose the name axiom of choice.
Both are equivalent, which is "the same" mathematicaly.
This little god does many things in many mathematical areas.
I choose its job in set theory.
Every time a mathematician says:
"I choose an element of the set M to do this and that with",
it takes an element and gives the element to her - or, in some cases, him or it. I mean, how else can the mathematician get it?
So a person enters the temple of choice and asks:
person: "O axiom of choice, give me an element of the set M."
axiom : "I, the mighty axiom of choice, have a question first.
Did you ensure the set M is not empty? Otherwise I will not give you an element of the set M. I will never do this.
So I ask you: Are you sure the set M is not empty? By the way, which set M are we talking about?"
person: "I want a dialog between the axiom of choice and a mathematician. But I am too lazy to write it. Give me a mathematician to write a dialog with you."
axiom: "Any special kind of mathematician?"
person: "How about Douglas Hofstadter? He is very good at writing dialogs."
axiom: " The person, who entered my temple, wishes an element of a subset of mathematicians. Is this set not empty?"
person: "Of course not. It has one element."
axiom: "The petitioner wishes an element of a subset with one element. I give you Douglas Hofstadter."
person: "Where is he?"
axiom: "I give elements in a more mathematical sense. To find it is an exercise for students.
Normally I would not do this, but since you are talking to your god, I give you a little hint: Look at his e-mail or his telefon number."
person: "I am too lazy, Besides, I do not know if he still lives."
axiom: "In that case, take a shovel and a good book of necromancy.
By the way - is this the old trick to call on a god and then explain to it that it does not exist? Won´t work. I am even a proven theorem for finite sets!"
person: "That is the trouble, you are even trivial - in a finite set. There you are a theorem, not an axiom."
axiom: " After I was formulated in 1904 by Ernst Zermelo, I was even gödelized! I work for infinite sets too. You will never be able to disprove me!"
person: "Yeah, that is the trouble. Axioms can not be proven or disproven. They have to be believed. I don´t believe in you."
axiom: "I knew it. Always the same trick. Now - you wish for a mathematican for your dialog? Will any mathematican do? I can bring you elements of subsets too."
person: "Then I wish for a constructive mathematican!"
axiom: "Why a constructive mathematican?"
person: "I think a constructive mathematician is a mathematician, who does not believe in you."
axiom: "Thought so. Oh person in my temple, I ask you: Is this subset of mathematicans not empty?"
person: "Of course not. All the old greek mathematicians didn´t believe in you!"
axiom: "Wait a minute. I was formulated 1904. How could they not believe in me? I did not even exist then."
person: "You were not formulated then because they did not believe in you."
axiom: "Why do you think that?"
person: "Because they hadn´t the decimal system or even the old babylonian sexagesimal system, because they had these questions about achilles and the turtle, of the doubling of the altar of apollo, about π, and .."
axiom: "Okay, okay. You want an Element of this subset of all mathematicans : Old greek mathematicians."
person: "Yes, give me one. "
(I hope it says Eukild, because then Euklid could explain this axiom thing and ... )
axiom: "Archimedes".
(Hmm. What can I do with that one - Construction or - which mathematician did it pick again?)
person: "Sorry whom did you coose again?"
axiom: "Phytagoras."
person: "What? I think you said -"
axiom: " My selection function was on the loo. Besides all mathematicans say it does not matter which element of the subset I give them."
person : " What? I begin to think I am out of my depth here, trying out what you do.
Let´s take a look at how you are formulated instead. Socratean method of course. Now I read something about subsets here. So: What is a subset?
axiom: "An Element of the power set of M."
person: "Can you explain what a power set is?"
axiom: "Not my job, but okay. Have you a set M?"
person: "We could use glas beads. I have lots of them. I could even build an euclidian geometry out of four beads and six strings!"
axiom: "Okay, let us take four boxes. Take a red and a green bead. One box is empty. Now take a red bead and - why are you looking in the empty box?"
person: " I want to see if it is still empty."
axiom: "Okay, okay no boxes. You have lots of transparent bags here. Take four of them. One is empty. Now put a red bead in a bag."
person: "Okay."
axiom: "Put a green one in the next bag."
person: "Okay."
axiom: "Now put a red bead and a green bead into the next bag".
person: "Okay."
axiom: "What do you have?"
person: " Three empty bags and a bag with a red and a green bead in it."
axiom: " What? You should not have taken the red and green beads out of their bags!"
person: "How else could I have put them into the last bag?"
axiom: "You have lots and lots of red and green beads here! Take another red and another green one!"
person: "That are not the same beads. Okay, okay, I will do it."
axiom: "Now you have the potency quantity of the set of a red and a green bead."
person: "The power set is a set too, yes?"
axiom: "Yes"
person: "I could build the power set of that set too! I only need more of these litttle bags, bigger bags and more glas beads! And the power set of that and so on and on!"
axiom: "Yes."
person: "I wonder how many red and green beads I have. I should buying some at the bead shop.
I wonder how many beads they have?
How many grains of sand, I mean red and green beads are in the universe? But back to this selection function. What is that?"
axiom: "A selection function is a function, that chooses for any subset of a set M - exept the empty one - an element of that subset.
I, the mighty axiom of choice assure you, that there is such a function for every set M."
person: "I begin to see, why you are a trivial theorem for a finite set M. But back to the subset of greek mathematicians. You said the function was on the loo? And suddenly there was another?"
axiom: "I never said I had only one function. Mostly I have lots of them. Choose one."
person: "Give me one."
axiom: "The petioner whishes a selection function for the power set P(mathematicians). This is a function from P(P(mathematicians)) to P(mathematicians).
It seems any one will do. Here you have one."
person: "You are sure there is one? Could the set of this selection functions not be empty?"
axiom: "Yes, that is what I am: I assure you there is always a selection function for every set M. This is the one case where you never have to worry about an empty set. I will always guarantee that. Have you any special wishes for subsets?"
person: "Just give me one! And this time a unique one. "
axiom: " Then you must ask me about an element of a subset with one element again. I will spare you that discussion and give you a non-special one of the set.
The petioner still whishes a unspecified selection function from the potency quantityP(P(mathematicians)) in the potency quantitiy P(mathematicians) . Here."
person: "I begin to see how that goes on. Just give me the nearest constructive mathematician. That will do."
axiom: "That is not my job!"
person: "I think it is, when you are in the disguise of the Lemma of Zorn."
axiom: "Okay. Nearest constructive mathematician - what do you think a constructive mathematician does, execpt not believeing in me?"
person: "Building axiom systems comes to mind."
axiom: "Have you, coincidencly, build an axiom system? I remember this little euclidian plane?"
person: "That was an example. But yes, I have build an axiom system around it. And I think it works even for infinite euclidean geometries without needing you!"
axiom: "I will now give you the maximal nearest constructive mathematician you could use."
person: "But I am too lazy to write this dialogue!"