# Power of the axiom of choice [closed]

Consider the different uses below

1. ”A person comes into a bar… ” – which I take to be a description of a joke which can be told without knowing the person’s actual name. This is as contrast to

2. A story about Obama, which might require the name to be presented.

I now wonder in a similar intuitive way (please take my ignorance into account) what mandate I get from using the axiom of choice.

Does it give me the right only to choose a number without actually knowing the value of the number, as in alternative 1, or does it, as in alternative 2. give me access to an expansion in a base, i.e. an infinite series which is available to me even if it requires an infinite instruction (and cannot be given in a finite form)?

Edit: I take my comment below as my edit to try to comply with the site´s rules for clarity:

@Mosher To rephrase my question to fit your function: Do you know, from the AC, only that such a function exist, or does the AC allow you to choose a particular case of this function - or do you perhaps consider these alternatives to be one and the same?

I see it as saying that a real number exist that you don´t know vs. allowing you to choose a particular number (in the form of a particular form of an infinite set) – or whether this is the same thing in your opinion?

## closed as unclear what you're asking by Andrés E. Caicedo, user147263, Mankind, Jeremy Rickard, drhabOct 10 '15 at 18:37

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• Can you be more specific. (1) and (2) have nothing to do with the axiom of choice. – Thomas Andrews Oct 10 '15 at 15:13

Instead, the mandate you get from the axiom of choice is the ability to choose a whole function's worth of elements. If you have a function $F$ defined on a set $I$ such that $F(i)$ is a nonempty set for each $i \in I$, then the axiom of choice tells you that there is a function $f$ defined on the same set $I$ such that $f(i) \in F(i)$ for all $i \in I$.
• This is not a philosophical or opinion based issue. This is an issue of how logic is used in mathematics. The basic principle of logic is something which I have heard called "choose it then use it", and it goes like this. Suppose you know have a statement of the following form which you know is true: "There exists $x$ such that $P(x)$ is true". Then, you may choose an element $x$, assume that $P(x)$ is true, and use the truth of $P(x)$ in your subsequent argument. – Lee Mosher Oct 11 '15 at 13:02