What exactly are the equations explaining why a singularity's mass is the way it is?
There are no equations "explaining" anything.
Assuming singularities exist (we don't know), their mass is simply given by the amount of stuff that fell into it. This is energy conservation.
But since it's density exists and it has no volume then it's mass should exist by multiplying the density by volume but then the mass would be undefined, so is there any other way to describe it that is not an empty set ∅≠(0).
The mass is not found by multiplying the density by the volume. You can't do it. That's what it's called a singularity. It's the point where the math stops working.
Density as a concept works for extended objects, but not for things with no volume like a singularity or a point particle. One does not talk about the density of a singularity just as one does not talk about the density of one single electron.
So the technically peace breaks math and physics, because infinity divided by 0 should give us zero but that doesn't work. So math breaks down at this point, thanks, that was the answer I was looking for.
Well, you can go from functions to distributions (more general structures compared to functions), where those things can get fixed in mathematics.
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