"Axiom" is just a fancy name for an assumption. Our view of the world is built on assumptions, and the logical structures that result from them. These assumptions cannot be proven, for even the principles that guide logical proof are also founded on unprovable assumptions.
We assume that the world is at least partially logical and structured. What does this mean?
In a partially logical world, we can point to some things that could, in principle, be contradicted. A historical claim is contradicted if the event never occurred. A house is painted or it isn't. A glass is broken or it isn't. 2 + 2 = 4 or it doesn't. Without logic, we would have to admit facts and their contradictions simultaneously. No proposition could be meaningfully true because its negation might also be true.
A partially structured world is one in which we can recognize some persistent patterns. 1 + 1 = 2 every time, not just the first time we compute it. A dropped glass falls to the floor every time. Most memories are recollections of past occurrences, not fantasies.
Life without these two assumptions is almost unthinkable, yet there is no proof at all without these axioms, so the axioms cannot prove themselves. Though we will be tempted to regard these axioms as necessary, they cannot be proved to be so. Any act of rational thought relies upon them, so they are necessary for rational thought, but does that mean we should accept these axioms? Can we imagine someone who rejects these axioms, and lives life in the moment, eschewing rational thought? It's difficult to imagine, but perhaps not inconceivable. Perhaps, if a person ignored all rational predictions, he might live in accordance with rejection of the two axioms. Not that I know of anyone who would actually do this. Logic and structure are too useful, too emotionally satisfying. For humans, it may be biologically impossible.
Though the axioms of logic and structure cannot be shown to be necessary in any broad sense, they are necessary for rational thought. If you're going to reason about philosophy, you have already accepted these axioms.
As it happens, these are the axioms of naturalism. These axioms predict that science will work. If we can find persistent logical structures in the world, then those structures make predictions. It's that simple.
Morality is a phenomenon that may end up being explained naturalistically. However, such an explanation only tells us why we feel to do what we feel we should do. It can only explain our feelings. It cannot tell us whether we should act on them or against them. In the eyes of naturalism, morality is as real as any other phenomenon, but it makes no prescriptions for how we should act. Naturalism can only tell us how we should act if we already know what goal we should seek, and it is that overarching goal that moralists want to identify.
What moral conclusions can the naturalist draw? Well, presumably the naturalist accepted the axioms of naturalism out of a desire to benefit from them. Humans naturally want to survive, to avoid pain, and to obtain pleasure. They can only act to achieve these things by assuming the axioms of naturalism. Otherwise, no action could be predicted to have desired effects. We each have our own thirsts to be quenched, and we're just using the predictability of the universe to assist us in quenching them. For the naturalist, the good is that which satisfies desire.
Where do the supernaturalists fit in? I think that most supernaturalists accept the axioms of logic and structure, making them baseline naturalists. However, they go beyond the naturalist's belief that we can find logically consistent systems, and beyond the belief that we can find structured things. Rather, the supernaturalist believes that the entire universe is logical, even when we cannot find the logic of it, and that the entire universe is structured even when we cannot find that structure. Supernaturalists may even propose that there are structures that are utterly undetectable.
These supernaturalist beliefs require additional assumptions, and these assumptions are motivated by new desires.
I think that morality is the primary motivation for accepting axioms. Morality, is all about how we should act. Naturalistic assumptions can tell us whether actions will lead to desirable results, but they cannot tell us what we should desire. For supernaturalists, this is inadequate. We know that people differ in what they desire. If another man's good is my evil, how do I know my good is, er, good?
Thus, the supernaturalist is drawn to make additional assumptions. Since naturalism cannot provide an "ought" from the observed "is," the supernaturalist proposes additional axioms:
1) There is an objective morality by which actions are judged to be right or wrong. This right or wrong is independent of personal opinion on the matter, and independent of social consensus. An act we consider evil, might actually be universally good, or vice versa.
2) There is a being that can see this objective morality. This is a huge assumption. What assumptions must the being adopt to be able to see this moral reality? What senses must the entity have? We cannot even imagine such a mechanism.
3) This moral sensitive can convey objective morality to humans. This is a relatively small assumption compared with the last two.
4) This being will convey objective morality so in a trustworthy manner. This assumes either a) that trustworthiness and truthfulness is a moral virtue and the being is morally good, or (b) that trustworthiness and truthfulness are evil but the being is objectively immoral. Another big assumption, and one that clearly begins to beg questions.
5) That such a being has communicated with us in the past. This sort of claim is rather different from normal testable claims. Objective morality is inherently untestable. All we have to go on are our personal feelings which have been assumed to be unreliable guides to objective morality. This is quite unlike a sighted man leading a blind man. A blind man knows when he is led into brick walls, or whether his leader appears to see objects at a distance. A morally led person has no way to know whether his leader is trustworthy.
Here we have five big assumptions. Why make them? When you ask the supernaturalist what's lacking with moral relativism, they will respond with a laundry list of moral transgressions, each one assumed to be allowable under the relativist worldview. Supernaturalists will argue that the murderer does what his desires tell him to do, so a murderer is no more wrong for himself than the volunteer firefighter is wrong for himself. I'll examine this claim below, but, suffice it to say, the supernaturalist desires to avoid a personal list of subjectively immoral acts. He wants to say that his rules apply to everyone, no matter what anyone feels.
This moral desire inspires the supernaturalist to adopt the additional axioms. It also inspires the supernaturalist in his decision on who to trust as an objective moral authority.
However, in taking this course, the supernaturalist reduces his worldview to a morally relative one. How can objective morality be arbitrated by a subjective decision to adopt unnecessary axioms, followed by a subjective decision to trust a source as authoritative and trustworthy? It cannot be so arbitrated. The supernaturalist merely cloaks his desires in objectivism to make moral choices look like logical deductions.
The saddest part of the story is that the supernaturalist chases this metaphysical goose for no practical gain. Our social structures of law and order form naturally from moral relativism. Objectivism is useless. Objectivism is not of any help when trying to persuade another party unless both parties have made the same assumptions. Thus, the supernaturalist's assumptions are pointless because they do not lead to avoidance of the supernaturalist's laundry list of evil activities. Like the moral relativist, the moral objectivist must still appeal to the desires of the offending party in order to persuade.
Thus, as far as morality is concerned, moral objectivists have no legitimate claim of superiority over moral relativists. Both have the same persuasive capability, and both are equally founded on personal desires.