Possible proof that the universe can not have infinite mass?

Question

I was thinking about the possibility of the universe having infinite mass, but then it occurred to me that it would then perhaps violate conservation of mass. Here is my thought process:

Assuming that "infinite" means an amount greater than any number, and

assuming that no mass can be created from nothing or become nothing,

this would mean that even if mass somehow disappeared from the universe, conservation of mass would not be violated due to the fact that $\infty - x$ is still $\infty$. This means that, at least mathematically, there wouldn't be a way to account for or "enforce" conservation of mass (or energy, either). Meaning that, the creation of any any mass from nothing and/or the disappearance of mass into nothing should theoretically not violate any conservation laws assuming infinite mass.

Is this a valid way to prove that the universe has finite mass? Does the universe have finite mass?

Regarding the possible duplicate question: The other answer doesn't quite suffice an answer to my specific question, in the sense that it addresses general relativity more than this specific "proof" or inquiry I had. If I hadn't received the current accepted answer, I don't imagine that I would have had my question answered in full.

Answer 1

In general relativity the conservation of mass, energy, and momentum are captured in the covariant expression $\nabla_{\nu}T^{\mu\nu}=0$. This is a local conservation law which basically says that energy, momentum, and mass cannot be created anywhere at any time. If you draw a small 4D box in spacetime any energy, momentum, or mass that comes in one side must go out some other side, although it need not be the opposite side.

This is what is meant by the conservation of mass. It is important to note that this is a local law which holds at every time and place in spacetime. However, because of curvature it does not necessarily hold globally. In fact, in some spacetimes it is not even possible to define a global mass, at least not in a coordinate-independent manner.

Because of this fact, any solution to general relativity, including the standard cosmological model, will necessarily obey the conservation of mass locally but may not have a well defined global mass at all. This is true both of cosmological models with positive curvature (finite) and cosmological models with non-positive curvature (infinite).

So the proof doesn’t work because local conservation of mass holds no matter what, regardless of whether the spacetime is finite or infinite, and global conservation of mass is undefined regardless of whether the spacetime is finite or infinite.