Imagine there's a sphere of radius $R$ which has constant density $\rho$ and you can stand anywhere inside of the sphere. Wherever you stand within the sphere, you'll only feel gravity coming from the mass at smaller radii than the point at which you're standing. That is, you could be standing at some point with radius $r$ and if somebody come along and added more mass to the sphere so that the radius was extended to $R'>R$ you would feel no effect because you're insensitive to it.

Puzzle: What happens if somebody comes by and adds mass to the sphere until it extends out to infinity? Now the mass distribution is uniform everywhere in space so you'd expect no gravitational pull in any direction, by symmetry. What happens? Do you suddenly stop feeling a force? If you have an issue with extending the mass to infinity then imagine that we've added a point to $\mathbb R^4$ so that we're on $S^4$.

What's the intuitive way to understand what's happening in this limit? What's the technical explanation? Is it a boundary value issue?

• Suggestion to the question (v1): Limit the scope of the question to Newtonian gravity in $\mathbb{R}^3$ or $(S^1)^3$ space for clarity. If you like this question you may also enjoy reading this, this, and this Phys.SE posts – Qmechanic Sep 7 '13 at 13:37