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Every fundamental force in our universe has a "coupling constant" associated with it that tells us about the relative strength and range of the force, compared to all others. For gravity, this constant is Newton's universal gravitation constant, G = 6.67x10-11 N-m2/kg2. In combination with other, known, universal constants (Planck's constant, the speed of light in empty space), it defines a length scale for quantum considerations in gravity. Putting these constants together gives the Planck length as about 10-35 m and a Planck time of about 10-27 sec.
This is much smaller than our present estimate of size for the typical radius of an atomic nucleus, or of a proton! Still, it is much different than "exactly zero." If particles, or for that matter the inherent structure of space and time, have a minimum size in this sense, then the problems with classically infinite values at zero size go away. In this view, nothing smaller than these minimum sizes can even be brought into question. There are no more geometric points to talk about. The currently unmeasurable radius of the electron can be no less than the Planck length, so its internal energy is finite rather than infinite. Indeed, space and time can be thought of as being "foamy" and the building blocks of the universe have a minimum geometric size in every way.
So, we need to take the universe from a different point of view, something that is inherently quantum or atomistic. Such theories are currently being very actively developed, debated, and re-written. A full background of this class of theories would be much more than a course in itself, but we will look at a few concepts of one of the contenders for the title - Theory of Everything! (Unfortunately, the concepts that we will deal with are beyond the range of our current technology to probe and test.)
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