This is my question. Black holes have infinite density right? (At least that is the leading theory), but if they have infinite density, and a finite amount of mass, then they must be infinitely small with means they can't have any volume. But that can't be possible because the definition of matter is anything with mass AND VOLUME. So that means black holes can't be made of matter because they are infinitely small, but they still some how have mass. I can't figure this out.
Well first of all the size of a black hole is defined by the event horizon. In fact the event horizon defines what a black hole is.
So we start from an event horizon, not from an infinitely dense point - that's not needed to form to a black hole.
We do need enough matter or energy (either or both) in a small region to form an event horizon. But that happens at a finite density. It does not have to be mass - just energy of any form.
The issue with "is there a singularity inside a black hole" is kind of tricky.
From outside the event horizon you have no way of knowing what's inside the event horizon. There's a concept called the no hair theorem which explains that you only get to know the mass, charge and rotational speed of the black hole as a complete unit. You can't know anything about the interior "structure".
So from the outside, the idea of "what is it like inside" has no useful meaning. From the outside the existence or not of a singularity makes no practical difference, in effect - the black hole will behave just the same with one as without.
Now inside the black hole, things are very, very different. All paths lead to the singularity in a classical black hole. Point anything at any speed in any direction and it will still end up at the singularity. And, from the point of view of someone inside the black hole, that will happen them in a finite time.
But inside and outside are effectively separate bits of the universe. They can't communicate and they don't experience the same things at all.
Now whether a singularity can really exist inside a black hole is a very debatable subject for theorists. To answer that question properly we would need a complete quantum theory of gravity - but we don't (at this time) have one. Best guess would be that the Heisenberg Uncertainty Principle would mean that at the core of the black hole we'd end up with a fuzzy region of uncertain make up and extremely high energy density, but not a real classical singularity, so (probably) no infinite density and no problem :-)
To be clear, black holes are objects with
- finite mass, and
- zero volume, as far as General Relativity is concerned.
This is subject to several nontrivial provisos, though.
- In general, we do not really expect the true core of black holes to be a mathematical singularity with zero volume; instead, we expect GR to fail at some point and to give way to some form of quantum gravity. However, that is thus far unknown physics and we don't know when those deviations occur or how they look like. As far as known physics goes, black holes do have zero volume.
- Not all mass is in matter, because the energetic content of any system also contributes to its mass through $E=mc^2$. Moreover, there is an energy cost to bending spacetime, and that also contributes to the mass of the black hole. (As a recent example, black hole mergers, as detected in LIGO, radiate away a significant fraction of their mass as gravitational waves.)
the definition of matter is anything with mass AND VOLUME
this is a made-up criterion with no support in any serious physics.