I think it depends on how God is defined.I guess in the end is just a matter of faith on both sides...
Take it to an extreme:
Suppose you have a being that appears before millions of people, performs feats that defy any conventional explanation which become documented, unexplained scientifically, and for all intent and purposes are "miracles". This being doesn't look like anything we conceive of as being "alive" for that matter, yet it's existing nonetheless in some fashion we have yet to identify.
This being makes a lot of claims: "I created all things. This entire universe, I created it. I'm responsible for many of the miracles you experience, though you can't fully explain it yet. Some of the claims of others are false, but not all of them," and this being goes on to further demonstrate even more amazing things. It then claims, "I am whom you refer to as God ... I am the only one."
On what basis do you accept the claims ? The demonstration of power ? Because you can't explain it ? Because you are overwhelmed ? How do you know the being is not lying ? After all, can you prove the being was the "first cause" ? How would you ? Can you prove there aren't any other beings like this one ? How would you do such a thing, especially if you only currently had "one" such being as a reference point anyways ?
So long as such a being is making ANY sort of claim to something which we cannot prove (i.e. first cause, only god, etc) then arguably no amount of displays of power, verbosity, etc will "prove" what that being is claiming. You will either trust the being, or you won't, for whatever reason. Some claims such a being may make, arguably, are accepted on trust and at face value.
Not all claims. Obviously such a being could say, "I can do such and such," or "I was responsible for such and such," and perhaps certain things can be verified. But there are arguably some claims that even a being that defies our ability to explain may make, which we would not be able to verify one way or another. If such a being made no such claims, and was solely about objective, falsifiable evidence ONLY ... that may be another matter. But so long as such a being could either make the claim that cannot be verified, or has such a claim attributed to it ... it would boil down to a person either trusting it, or not.
Thus, one could argue there is no point in such a being claiming such things one way or another, or even clarifying them for that matter ... not if the point is to "prove".
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