First, I think you mess up terminology somewhat. To be more precise, axioms and postulates. An axiom should be treated mostly as a basis for abstract analysis. According to Kant, analysis doesn't give any essentially new data, it just reveals the "hidden" properties of the initial statement. Well, an axiom is that initial statement. The difference from postulate, however, is that that initial statement may be totally random in case of an axiom, since it's absract. I can axiomatize "10=0" (oh well, we all know I actually can't, since I will also need to redefine "10", "0" and "=" in a different way that doesn't automatically imply that "10!=0" which it does in common definitions, but that's not the point, let's take it the simplest) and build a mathematical theory out of it, and it will be valid. To be more precise, validity is not even a concern here, as it's not something that can be applied to axioms. In contrast, a postulate is a statement that too, serves as a basis for analysis, but actually comes from some observations of the real world. It's not necessary to delve deeper into this, it's enough to say that postulates are based on observations, while axioms are based on nothing. Therefore, it's perfectly okay to question the validity of a postulate, but not an axiom. That for Neumann thing is somewhat different, as it's just an assumption that was later crushed by analysis, it's not an axiom, but since it's on the borderline between empyric and abstract (as much of probability and statistics is), it's best called a hypothesis, much like the ergodic hypothesis. Nothing wrong with a hypothesis proven false.
Well, that was somewhat offtopic, but messing up definitions when talking on such complex subjects generally leads to nothing good. Returning to the subject of faith (again, a little more proper word for this case, but not nearly that relevant), speaking of it, it's quite logical to discuss god. I'm sorry if my wording is poor, it's hard to discuss it even in my native language. God is an absolution of the idea of faith and studying extreme cases of something is a common scientific method. Well, as an idea, what properties does god have? It's main property is absolution, let's assume it takes manifest in absolute power in context of the material world. Definition is the hardest thing here, it will probably lead to a different result if defined differently, but I'll also try to provide some basis for why I define it like this. Had power not been absolute, the subject should be adhering to some sort of law, physical, mystical or whatever. Something that strictly adheres a law, however, is a phenomenon. Now moving on to the next part - to scientifically prove if something exists, we must conduct an experiment. What is experiment in it's essence? In essence, an experiment is a question to nature, to the world, but the point is that it's a question nature has to answer. If me measure voltage, we may ask "is voltage above 100mV?" and there is no way that question won't be answered. You can't get "no answer" from nature when experimenting, unless, of course, your equipment stops to function (but as fun as this picture is when applied to a god-measuring experiment, equipment malfunction in 100 consecutive tests is a result as well). One of the main postulates of quantum mechanics is that it's impossible to conduct a measurement that won't influence the object being studied. Experiment, therefore, implies we take some action on the studied object it can't resist. So, assuming it's possible to conduct an experiment that would prove the existence of god, we're automatically saying that we can ask god questions he can't refuse to answer or, wording it differently, we can apply some sort of force to god, which contradicts his absolute power. Therefore, in we define god with absolution in mind, it's inherently immeasurable. Inherentely unobservable. That doesn't in any way "prove" that god doesn't exist, since any kind of that proof would too rely on an experiment, but it immediately raises a point with the neo-positivist "verifying principle" that states:
Whatever isn't verifiable (it's impossible to think of an experiment that proves or disproves our statement) is not a scientific matter.
That simple, absolute god is not a scientific matter. The only way it limits us that any attempts to "prove" god's existence are fool's endeavours and frankly I'm amazed how many pseudo-scientifical books came out lately that try to blend god and science, and even more amazed at how many people actually buy this bullshit.
We can, however, define god as a non-absolute something, but those definitions are all inherently flawed. They are all either taken out by Occam's Razor (if god is just something that designed human without any additional properties, well, isn't it easier to assume that it's just an extraterrestrial intelligence, besides, synergetics tell us that simple laws with large numbers may lead to complexity and order being formed out of chaos by itself which is even more preferable) or have inherent definition loops (god made us, who made god?). The presence of absolution in god's definition should seem obvious.
Well, that was a long introduction for a very short point. Faith is irrational. It doesn't have any basis and can't have one. You just believe in god, you draw him in whatever colors you want, and it's meant to make you happier. Nothing more, leave the explanation of the universe to science. Besides, science doesn't answer the question "why?", it only answers "how?", so the former one is still a matter of faith.
Finishing the post with some offtopic, I'll quote one post from this thread:
"If atheism is a religion then not collecting stamps is a hobby."
This is both right and wrong. From one point of view, it's not a religion because it doesn't have the religious traits - no rites, no sacred texts, no clergy. But from another point of view, it IS a religion because believing god doesn't exist is just as irrational as believing it does. Agnosticism seems much more of a rational position.
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I apologize to all people who might be offended by me referencing god as "it", but I'm more talking about god as an idea, without any gender specification.