Why is the axiom of choice separated from the other axioms?
Solution 1:
This is a historical issue really.
Originally set theory was developed by Cantor and the well ordering principle was somewhat assumed in the background (e.g. Cantor's proof of the Cantor-Bernstein theorem was a corollary from the fact that every two cardinalities are comparable).
In 1904 Zermelo formulated the axiom of choice and proved its equivalence to the well ordering principle. He later formulated more axioms which described our intuition about sets, therefore removing the "naivity" from the Cantorian set theory. He did not add the axiom of foundations, nor the schema of replacement. Those were the result of Skolem and Fraenkel which were popularized by von Neumann.
The axiom of choice remained controversial, the thought that the continuum can be well-ordered was mind boggling and caused many people feel uneasy about this axiom. Further results like the Banach-Tarski paradox did not help to accept this axiom either.
Prior to set theory most mathematics was somewhat constructive in the sense that things were finitely generated or approximated by finitary means (e.g. limits of sequences). It requires quite the leap of faith to go from things you can pretty much write down to things which you cannot describe but only prove their existence. In this sense the axiom of choice augments the way we do mathematics by allowing us to discuss objects which we cannot describe in full.
It was questionable, therefore, whether this axiom is even consistent with the rest of the axioms of set theory. Gödel proved this consistency in the late 1940's while Cohen proved the consistency of its negation in the 1960's (it is important to remark that if we allow non-set elements to exist then Fraenkel already proved these things in the 1930's).
Nowadays it is considered normal to assume the axiom of choice, but there are natural situations in which one would like to remove it or find himself in universes where the axiom of choice does not hold. This makes questions like "How much choice is needed here?" important for these contexts.
Some things to read:
- Why worry about the axiom of choice? (MathOverflow)
- Advantage of accepting the axiom of choice
- Axiom of choice - to use or not to use
- Foundation for analysis without axiom of choice?
Solution 2:
In ZFC, there are three particular axioms that are less obvious than the others: regularity, replacement, and choice. (Replacement is an axiom scheme, but we can ignore that difference for this purpose).
Of these, regularity (well foundedness of $\in$) is the easiest to deal with. Although there is no reason to think that our naive conception of sets eliminates the possibility that there is a set which is a member of itself, it also turns out that we essentially never construct such sets in the course of ordinary mathematics. Thus the axiom of regularity does little harm (in removing things that we care about). It does do some good, as well-founded models of set theory are much more convenient to study. Most mathematicians never think about it.
The axiom of replacement is odd because it is hard to motivate directly from the notion of the cumulative hierarchy; replacement is essentially about the length of the ordinals rather than about which sets exist at each level of the cumulative hierarchy. There are very few mathematical arguments outside of set theory that actually use this axiom, though. The main examples are the Borel determinacy theorem and some theorems from category theory. Thus most mathematicians rarely notice it, it is not mentioned in many undergraduate books outside set theory, and except for set theorists I expect few would be able to state it without thought.
The axiom of choice is odd because it is a set-existence principle, but as t.b. says in a comment it is not implied by the other set-existence scheme in ZFC, the separation scheme. Unlike replacement, though, the axiom of choice can be motivated from the naive construction of the cumulative hierarchy. Historically, the axiom of choice was a flashpoint for certain discussions about constructiveness in mathematics, and for this reason, many authors in the past marked results that used the axiom of choice so that it was clear when it was used. This habit has decreased over time as the arguments from the early 20th century have faded somewhat into history; a side-effect of the habit is that it reinforced the lingering idea that there was something unique about the axiom of choice compared to the rest of ZFC.
All three of these axioms (regularity, replacement, choice) are, at various times, separated off from the rest of ZFC, leaving behind weaker set theories. The main reason that people think of the axiom of choice as special, rather than regularity or replacement, is that the axiom of choice has been the one that is most discussed in popularizations and undergraduate textbooks. But from the point of view of ZFC it is not at all the only axiom that requires effort to motivate.
Solution 3:
The basic axiom of "naive set theory" is general comprehension: For any property $P$, you may form the set consisting of all elements satisfying $P$. Russell's paradox shows that general comprehension is inconsistent, so you need to break it down into more restricted types of comprehension.
The other axioms of ZF (except for well-foundedness) are all special cases of general comprehension. For example, the Power Set axiom asserts that the class of all subsets of $X$ is a set. Replacement with respect $\phi(x,y)$ asserts that the class of pairs $(x,y)$ satisfying $\phi(x,y)$ is a set. Separation asserts is obviously a sub-case of general comprehension.
Choice is very different, because it asserts the existence of a set which does not satisfy a specific defining sentence.