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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?

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

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ Can you be more specific. (1) and (2) have nothing to do with the axiom of choice. $\endgroup$ – Thomas Andrews Oct 10 '15 at 15:13
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The axiom of choice is not about choosing a single element: if you know that the population of the bar is nonempty, you may choose an element without using the axiom of choice.

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 $i \in I$.

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  • $\begingroup$ @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. $\endgroup$ – Mikael Jensen Oct 11 '15 at 7:34
  • $\begingroup$ 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. $\endgroup$ – Lee Mosher Oct 11 '15 at 13:02
  • $\begingroup$ I am at the moment more interested in the chosen x but I take it that you consider that I can choose an actual value x. $\endgroup$ – Mikael Jensen Oct 11 '15 at 17:10

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