If someone were to say to you, "prove that God can't possibly exist", wouldn't you be tempted to say in response: "I can't prove how it is impossible unless I have some idea of how someone might think that it is possible. How do you think that it is possible for a God to exist?"
Actually, I'd be tempted to say, "No"

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Thinking about it, I wouldn't say what you said. God, however defined, is automatically possible unless proven impossible - otherwise, we're making unjustified
a priori assumptions. Given the definition of 'God', once might then be able to show that he can't, in fact, exist, but the default is that his existence is possible, not that it's impossible.
I find myself in just that situation. You suggest that it is inevitable that something would come into existence from "nothingness", but what is this process of something coming into existence? How am I to make sense of that?
You aren't, as it's not a process - that implies something material changing over time, neither of which is true.
I'm suggesting that, without anything to prevent it from happening, a thing
can happen spontaneously, and, in the absence of time,
will happen immediately.
I don't really have a problem with such randomness, though I don't think it is causeless except according to certain models of causality.
I don't understand. It's not causeless, unless it's causeless?
Perhaps "the universe" or "the fabric of spacetime" has the causative power to generate genuinely unpredictable phenomena, but this wouldn't be the same sort of causation as the "deterministic" sorts.
Genuinely unpredictable phenomena of the deterministic sort?
I ask this because an "event" is often thought of (in classical Newtonian mechanics) as caused by a prior "event".
Yes, well, so much for Newton
In any case, I would simply ask you: "how do you explain causeless events? Why are there causeless events and not no causeless events? Don't you think that begs for an explanation?"
Maybe, but the OP asks you to prove they aren't; it makes no claims that they are. But if these are causeless events, they wouldn't
have an explanation, not in terms of, "Oh, it was this thing here that caused it to happen".
Then we have a problem, since I don't believe that time exists outside of physical reality. So, it is senseless to me to think of there ever being a "time" in which physical reality did not exist.
That doesn't require that physical reality has an infinite past, but simply that its "beginning" just isn't like the beginning of any entity within the context of the universe.
Anyway, since I can treat all of physical reality as an "object", I don't think that I am willing to say that it is impossible for it to exist without a beginning as you have defined it.
Score one for me
