# Is the mass of the universe finite (collection of objects)?

The observable universe has a finite volume.
Density is (locally) the ratio of mass to volume.
For the mass of the universe to be infinite, the local density of the universe must be (somewhere) infinite.
At the location where infinite density is to be found, the curvature of space-time (i.e. gravity) must also be infinite.
Since the space-time curvature follows an inverse-square law, if there is an infinite curvature *anywhere* there must be an infinite curvature *everywhere*.
[Infinity]/R^2 = [Infinity] for all real values of R.
There is a place where space-time curvature is finite.
Therefore space-time curvature is finite *everywhere*.

PeterDonis
Mentor
2020 Award
For the mass of the universe to be infinite, the local density of the universe must be (somewhere) infinite.

Nobody is claiming that the mass of the observable universe is infinite.

space-time curvature follows an inverse-square law

No, it doesn't. Don't confuse Newtonian gravity with GR.

The observable universe has a finite volume.
Density is (locally) the ratio of mass to volume.
For the mass of the universe to be infinite, the local density of the universe must be (somewhere) infinite.
At the location where infinite density is to be found, the curvature of space-time (i.e. gravity) must also be infinite.
Since the space-time curvature follows an inverse-square law, if there is an infinite curvature *anywhere* there must be an infinite curvature *everywhere*.
[Infinity]/R^2 = [Infinity] for all real values of R.
There is a place where space-time curvature is finite.
Therefore space-time curvature is finite *everywhere*.

Do I detect a whiff of non sequitur?
I might assume that you have a cogent support for your first statement, though it seems to assume that the infinite mass of the universe is of the same or greater order than the volume of the universe. If the volume is of a greater order, you will have to explain why the infinity of mass must fill it. If the volume is in fact of the same order and its average density is everywhere the same on the same scale as we observe, then it would have infinite mass of the same order as the volume of space, without anywhere having infinite density (except possibly in some black holes somewhere FAIK).
Density is not mass, nor vice versa.