I have always held that there are different levels of belief.
For example believing in the Axiom of choice requires less effort than believing in God, although neither are provable.
One appeals to my logical faculties and another might appeal to my emotional faculties.
Is there any way to say one belief is "stronger" than the other? Certainly one might be more useful than the other, but what are the commonly held arguments for these degrees of beliefs – whether for or against?
I have always held that there are different levels of belief.
It really depends on what you mean by "stronger". Your question seems to be looking for a way to make two very different concepts of belief somehow comparable. There are a few ways to go about looking at that, but we first need to get clear on the two different versions of "belief" in your question.
You "believe" axioms for the sake of argument. Let's call this belief(1). You generally "believe" in God in a different way - you don't just accept God's existence for the sake of argument, you accept it as an impersonal reality. We'll call this belief(2). Belief(1) is systems dependent - when you move from Euclidean to non-Euclidean geometry, the axioms change. They haven't been falsified, but largely because you also never made a truth claim about them. You simply said that these assumptions look very reasonable and we need them to do interesting things with bearing on reality, so we'll treat them as though they were true, sidestepping altogether the question of whether they really are true.
Belief(2) is different. You don't just treat the claim as true for the purposes of doing something else. Belief(2) is an impersonal, systems independent truth claim. It's more impactful in that you're not just treating the belief as the foundation for an extended thought experiment. In some ways, this makes it more sturdy, in others, less.
Now, as I say, there are a few different ways of comparing these two to see which is stronger, and it depends what you mean by "stronger." I'll give three, but I don't claim this as an exhaustive listing by any stretch of the imagination.
First, we could consider the capacity of each for impact. In this respect, belief(2) is pretty clearly stronger than belief(1). Because belief(1) is systems dependent, it has no real facility to reach beyond the system into which it is bound. It has impact on the (intellectual) world only so long as its system is widely accepted. Belief (1) may give us new ways of looking at things, but it ultimately makes no claims about the way things really are. As such, it has relatively little power to change the world. This is not to say that its power to change is small, merely that by comparison to belief(2), belief(1) generally won't change as much as fast or for as long.
Closely tied to that first method of comparison is a second, the commitment each requires from adherents. Belief(1) is easily abandoned. Once the system to which it is tied falls out of favor or ceases to be useful for a current project, belief(1) can be left behind quickly and easily. Students learn to do this as their intellectual life gains complexity and they discover that principles which once seemed sure bedrock are actually more like sand traps.
Belief(2), by contrast, requires significant commitment. It requires more intellectual and moral investment, more time to create, and more time to destroy. It would be appropriately cognized, I think, under the name "worldview", in this respect, and many people have belief(2) about things like religion or politics. There's a reason civil discourse steers clear of those two items in the interests of remaining civil.
The contrast here may be nicely brought out by considering what it might look like to have belief(1) concerning one's political views. Consider, for instance, the explanation of a senior civil servant to a junior civil servant concerning the nature of his beliefs: "I have served eleven governments in the past thirty years. If I had believed in all their policies, I would have been passionately committed to keeping out of the Common Market, and passionately committed to going into it. I would have been utterly convinced of the rightness of nationalising steel. And of denationalising it. And renationalising it. On capital punishment, I'd have been a fervent retentionist and an ardent abolishionist. I would've been a Keynesian and a Friedmanite, a grammar school preserver and destroyer, a nationalisation freak and a privatisation maniac; but above all, I would have been a stark, staring, raving schizophrenic." (Yes, Minister, "The Whiskey Priest"). The senior civil servant very clearly holds his political views as belief(1) in the interests of being able to do his job. Most people are not so flexible.
The third way we might compare the two is in fallibility, and here, belief(1) shines. Because it's not making a truth claim at all, just presenting a hypothetical set of principles for the world to see what comes of it, belief(1) is basically unfalsifiable. Belief(2), on the other hand, is making a truth claim, and usually a very strong one. It therefore can be falsified. Belief(1) is the less prone to being wrong, but largely because it doesn't try to be right.
All of which is to say, you've asked a very complicated question to which no simple answer exists.
I am really surprised. Believing in the axiom of choice means that you believe in what has been proven impossible, namely to well-order the real numbers or any uncountable set.
In order to well-order the elements of a set you have to distinguish them. That cannot be accomplished other than by names. But there are only countably many names. This has already been understood by Brouwer.
Brouwer, in his dissertation, refutes the well-ordering theorem by pointing out that in the case of the continuum most of the elements are unknown, and hence cannot be ordered individually – "So this matter also turns out to be illusory." (Thesis p. 153) Examples of (according to Brouwer) meaningless word play are the second number class and the higher power sets. [Dirk van Dalen: "Mystic, geometer, and intuitionist: The Life of L.E.J. Brouwer", Oxford University Press (2002)]
Therefore the axiom is as counterfactul as the axiom: You can find 10 different natural numbers less than 5. Of course, in a system having this axiom you can claim the existence. But you cannot really find these numbers. The same is true with the axiom of choice.
Cantor and Zermelo were of the opinion that every set can be well-ordered. Zermelo axiomized this belief in his axiom. He meant that this can really be accomplished. His paper is entitled: Proof that every set can be well-ordered ("can be" - not "has some well-order" or so). But he was in error. This error is not only manifested by Brouwer's argument, but also in pure logic it can be proven and has been proven:
No set-theoretically definable well-ordering of the continuum can be proved to exist from the Zermelo-Fraenkel axioms together with the axiom of choice and the generalized continuum hypothesis. [S. Feferman: "Some applications of the notions of forcing and generic sets", Talk at the International Symposium on the Theory of Models, Berkeley (1963)]
For example, it is a theorem that there does not exist any way to ever actually construct or even define a well-ordering of the real numbers. [William P. Thurston: "On proof and progress in mathematics", arXiv (1994) p. 10]
So the belief in the axiom of choice is comparable with the belief in something that with absolute certainty cannot be accomplished, like the axiom: In Euclidean space a straight line can be found to cover every triple of points. The belief in this axiom is as justified as the belief in the axiom of choice.
The axiom of choice would have been disposed of long ago - at least after Feferman's proof - but with it large parts of set theory would vanish - and that has to be prevented by all means.
If you see only a 1 % chance for the existence of God, then I would strongly recommend to revise your point of view. The belief in God is overwhelmingly more reasonable than the belief in the axiom of choice. His existence has not yet been disproved.