For those of us like myself who are terribly naive, why wouldn't that just cause all of the atoms to be torn apart?
Unless the sphere were infinitely far away, only the center point would be in balance, and everything else would be immediately sucked to the edge of the sphere. If the sphere's radius was infinite, that might explain the expansion of the universe itself, but not the presence of infinite density within the universe itself.
When you do the math for gravity inside a shell of material, the quadratic falloff exactly counteracts the distances and the total pull is exactly zero.
Why? At a distance, gravity doesn't really care about density, only mass.
An infinitely dense object would shred anything it touches by its infinitely high tidal force, but there's only a limited amount of material it can touch within a given time and the Universe is not infinitely old.
> but there's only a limited amount of material it can touch within a given time and the Universe is not infinitely old
Okay, to be precise something like that can't have existed in the observable universe (at the time we observe), but could exist in a distant bubble that's expanding at the speed of light.
There would be infinite m/v there, and finite everywhere without infinite density, so literally every single random draw of a single mass-unit would be in an area of infinite density, or so goes the idea.
That's presupposing that the mass portion of m/v is infinite, which, from our understanding of black holes, we have no reason to believe. Pathological local densities tell you nothing about the mass distribution of the universe precisely because they're so pathological.
Every theory has a domain of applicability, determined by tests within that domain. I don't think its reasonable to assume that the domain of applicability of any our theories extends to various infinities.