TITLE: Axiom of Choice: What exactly is a choice, and when and why is it needed?
QUESTION [27 upvotes]: I'm having trouble understanding the necessity of the Axiom of Choice. Given a set of non-empty subsets, what is the necessity of a function that picks out one element from each of those subsets? For instance, take a look at following paragraph from Wikipedia:

A sketch of the proof of Zorn's lemma follows, assuming the axiom of
  choice. Suppose the lemma is false. Then there exists a partially
  ordered set, or poset, P such that every totally ordered subset has an
  upper bound, and every element has a bigger one. For every totally
  ordered subset T we may then define a bigger element b(T), because T
  has an upper bound, and that upper bound has a bigger element. To
  actually define the function b, we need to employ the axiom of choice.
  Using the function b, we are going to define elements a0 < a1 < a2 <
  a3 < ... in P. [...]

To me it is clear that the ai's exist so why do we need singling them out? Sure enough, the binary relation representing b may not exist as a set, but what do we need it for?
Finally... if we actually need the Axiom of Choice to single out the ai's in the above paragraph, since the axiom only postulates the existence of choice functions, how do we single out the choice function itself so that we can select the ai's with it? Certainly not with another choice function, since we would need to choose this function by a third choice function.
So when and why do we really need the Axiom of Choice?

REPLY [3 votes]: Edward Nelson explained this very well in Chapter 2 of a paper that used to be on his homepage at Princeton ("removed for revision" several years ago). I'll try to do same using my understanding gained from it since I can't point you to the paper anymore. In logic it is shown (extension by adjunction of a function symbol) that if $\mathbf{\varphi}$ is a formula whose free variables are $\mathbf{x, x_1, \dots, x_n}$, and if $\mathbf{\forall x_1 \dots \forall x_n \exists x \varphi}$ is a theorem, then we can adjoin a new $n$-ary function symbol $\mathbf{f}$ and the axiom $\mathbf{\forall x_1 \dots \forall x_n [fx_1 \dots x_n/x] \varphi}$ to get a conservative extension. But such a conservative extension if applied to set theory does not contain new set theory axioms allowing the new function symbol to appear in the axioms obtained from the separation (subset) and replacement (or collection) schema. In practice, this makes no difference if the extension is what is called an extension by definition, which happens when the stronger statement $\mathbf{\forall x_1 \dots \forall x_n \exists ! x \varphi}$ holds (where $\mathbf{\exists ! x}$ means "there exists a unique $\mathbf{x}$ such that"). The reason is that in this case any formula containing $\mathbf{f}$ can be translated into an equivalent (in the extension) formula not containing $\mathbf{f}$, which formula is permissible in separation and replacement axioms. Also, when $\mathbf{\varphi}$ contains no free variables other than $\mathbf{x}$ there is no problem with allowing the (special constant, $0$-ary) $\mathbf{f}$ to appear in the replacement and separation axioms, since such would-be axioms would hold when $\mathbf{f}$ is replaced by a variable not appearing in the would-be axioms (and substitution holds).
Similarly, if $\mathbf{\psi}$ is a formula with free variables $\mathbf{x_1, \dots, x_n}$, we can adjoin a new $n$-ary predicate symbol $\mathbf{p}$, along with the defining axiom $\mathbf{\forall x_1 \dots \forall x_n (p x_1 \dots x_n \leftrightarrow \psi)}$. These also are called extensions by definition, presumably because statements involving $\mathbf{p}$ are translatable to equivalent statements not involving $\mathbf{p}$. Being translatable, there is again no problem allowing new predicate symbols arising from extensions by definition of a predicate symbol to appear in separation and replacement axioms.
In practice, in doing set theory or the higher math based on set theory, one may use extensions by definition and extensions by special constants with abandon, even in separation and replacement axioms. Indeed, math would be impractically tedious unless this or something similar were done. But whenever one uses extensions by adjunction of a function symbol which are not extensions by definition of a function symbol, as happens when the uniqueness condition fails, one needs something like the axiom of choice to reformulate those function symbols so they may be used in separation and replacement axioms. In practice, what this means is that any such function symbols should correspond to functions obtainable by the axiom of choice. Often (e.g., when variables range over sets like the set of real numbers) the axiom of choice can be used to replace something like $\mathbf{\forall x_1 \dots \forall x_n \exists x \varphi}$ with something like $\mathbf{\exists f \forall x_1 \dots \forall x_n [f(x_1, \dots, x_n)/x] \varphi}$, where here $\mathbf{f}$ is a variable representing a set-theoretic function and the parentheses denote a function symbol defining evaluation (to every function and argument there corresponds a unique evaluation of the function at that argument, and so evaluation may be defined problems-free by extension by definition of function symbol). The statement $\mathbf{\exists f \forall x_1 \dots \forall x_n [f(x_1, \dots, x_n)/x] \varphi}$ can be used to adjoin a function symbol $\mathbf{f_c}$ corresponding to $\mathbf{f}$ that would cause no problems appearing in separation or replacement axioms whenever $\mathbf{\varphi}$ involves only $\mathbf{x, x_1, \dots, x_n}$ as free variables, because then $\mathbf{f_c}$ would be defined as a special constant (confusingly, $\mathbf{f_c}$ needs to be $0$-ary).
Bourbaki, by using Hilbert's tau operator (which I guess is similar or the same as Hilbert's epsilon operator) does something similar to allowing formulas obtained by mere adjunction of function symbols to appear in separation and replacement axioms (both approaches would cause the axiom of choice to be a theorem), but as much as I like Bourbaki, I have come to prefer here the standard way logicians do things, which uses an axiom of choice. Assuming too much is not aesthetically pleasant.