In most mythologies a divine being is endowed with several innate characteristics. I want to highlight some and discuss how they relate to what we know.

Math

A divine being is usually tasked with judging the souls and lives of their followers or people in general. This task needs a rule set by which the person is evaluated. Image for example that everything a person is can be converted into some numerical or analytical representation and then absolutely logically judged.

The issue with this is that a complete consistent and fair rule set, as expected of a divine being, will have cases where the some statements will be false or true but without the possibility of a proof within the rule set itself. In addition, the consistency of the rules cannot be proven within the same system (see Gödel's Incompleteness Theorem for both statements).

In a sense this means judgement will need to have some additional divine logic outside the rule set itself, effectively making the judgement incomprehensible to the judged person. Which is completely fine, as even the rule set itself would be so vast and complicated to be essentially non-comprehensible.

On the other hand, a divine being will not be able to create a fully comprehensible rule set for their followers, effectively showing the limits of a divine being. Because of this, the question of what is divinity itself is raised. Is it a very powerful being divine or is divinity only found in the absolutes of infinities?

Physics

A similar argument can be made in physics about the changes a divine being can make in the physical universe. Any action taken by a divine being could be measured by people with good enough detectors. For example detecting a violation of a conservation law. After excluding enough theories, only the last resort remains, i.e. some outside force did it. This would prove that a divine being was manipulating reality.

Here, the same argument can be made again of the matter of degree of divinity. If the being needs to use physical space, then there might be limits on the scope of their powers. Even if the scope is the whole universe, the fact that there must be some physical action leads to the possibility of reverse engineering it, or blocking it, with enough understanding. This would effectively limit or mimic divine power.

There are several outs from this discussion. One is that the manipulation by the divine being is on such a fundamental level, that no detector will ever find it. For example on some kind of hidden unmeasurable space time parameters. But in this case, as the manipulation is complete, the old question of free will and determinism emerges naturally.

The other out is that the divine being set the universe up to just observe. In this case there are two general believe sets: The first, that the universe just does its thing, with no morality and more or less only guided but what emerges, or secondly, the perfect setup with a plan that flawlessly plays out. Both will be unknowable from the perspective inside of it.

Information

Another interesting point is the transfer of knowledge from a divine being to a human. If it is some kind of transmission the same argument as before applies and it should be detectable.

If the information is not physically transmitted, the only knowledge available would be the one already kept by the human. This means that any word-of-god would just be the experiences, the thoughts and knowledge laundered by a believe of divinity. This leads to basically declaring whatever someone thinks as divine rules.

Actual divine knowledge will be either worked out by the scientific method or by philosophical and societal work. Anything else is just a snapshot of what someone thought at the time.

There is an argument to be made, that this is exactly what the divine being wanted by picking a specific person with the right background and somehow triggering divine inspiration. But the questions arises, why this messy process? Instead, for example a more obvious transfer of knowledge like carving some messages into stone observed by many people at once. The only out for this is, that the divine being wants to be messy and ambiguous for whatever reason. Or is just not capable of a clear message either because it has not the power to do so, or because it cannot express its thoughts simply enough for people. Both are limits to its divinity.

Randomness and Patterns

A classical argument for the existence of a divine being is the interpretation of events, either small or big, to be some form of intervention. Modern evidence for this intervention are usually based on the chance of something happening, and deciding that if something is unlikely to happen, that this must be evidence of a higher power.

This can be seen as a natural outcome of the human mind, which is looking for patterns in everything. And if it is primed to believe in a higher power, than everything becomes an expression of that. On the other hand, most critics will cite exactly this, a random event is misunderstood due to seeking out some meaning in the chaos of life.

This leads to the question, why divine intervention is not more obvious. Is it because the higher power wants people to be open to believe in it or otherwise choose to ignore it? Why would it choose this? To allow for some free will? Or is it because it only observes?

In general, I think believe systems emerge due to a need for an explanation of nature, good and bad events, or just to have an authority to enforce a human created rule set. Without actual scientifically proven evidence (and hence divine limits), any believe system like this will function as a bridge to the divine, in most cases as a foundation for everyday life, a feeling of to be taken care of and a form of guidance.

In a sense it does not matter exactly what the divine being is, its rules or anything else. As long as it stays hidden it fulfills its role in human society and people will fill gaps as much as they need to. One can even argue, that with the presented arguments, it cannot show itself properly without the risk of dispelling its divinity.