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I wonder if one can define the axiom of choice by the assumption that is possible to choose an arbitrary real number?

If we define real numbers as Cauchy limits of rational numbers, I assume that we must consider that as a set, since it is a case of so called actual infinity, referencing directly an infinite number of limits.

In any case that would just mean choosing one element out of one set, rather than involving an infinite number of sets, in what you see in the formal definitions.

My question is therefore: is the formal – complicated - definition absolutely necessary?

Edit: Based on the answer by Asaf Karagila I realize I was hasty with the reals. I expect I can mend it, either by accepting the equivalent classes of limits of Cauchy series, or by restricting the series by using infinite (decimal) expansions and the added rule that series of 1000….has priority over 0999… (these series containing 0999… are then discarded ) which I believe would make the definition unique.

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You are being too vague. The real numbers is a specific set. Is it the set of Cauchy sequences modulo some equivalence relation? Is it the set of Dedekind-cuts?

In any case choosing a real number, is choosing an element from a set. A non-empty set (assuming you accept the rational numbers exist).

Choosing one element from a provably non-empty set requires no appeal to the axiom of choice, but rather an appeal to the laws of your logic. Specifically, existential instantiation. If $\exists x\varphi(x)$ is provable; then we can add a new constant symbol to the language $c$ and the statement $\varphi(c)$.

Since we can prove the real numbers form a non-empty set, we can always choose one.

The fact that you chose to define your real numbers as equivalence classes of Cauchy sequences is meaningless. You choose a sequence. The sequence itself is an object on its own accord. And you didn't have to "choose its elements". Once you posited that you consider all the sequences with such and such properties, all those sequences gather into your set.

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  • $\begingroup$ I thought (perhaps in error) that choosing was equal to defining an element among uncountable elements, such as the pointwise definition of real numbers. I remember now that you - and Joel Hamkins - has shown that such a definition is possible using inter alia the axiom of choice. Since there are many intermediate steps in the proof, I realize the AC cannot be equal to this definition. It seems that you think such a choice possible, or is choosing not equal to defining, because I can’t see how you can define a real number without an infinite sequence of instructions (without invoking the AC). $\endgroup$ – Mikael Jensen Jan 10 '17 at 10:11

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