So I was in the shower today...

[Zack]

Zack: In calculus, we use 1/0 = infinity all the time. It's one of the fundamental axioms which allow us to determine the limit of an equation.

Agamemnus: Gravity on any certain point will be equal to the summation of the product of each mass surrounding it divided by R^2, with a scaling factor thrown in there. This does mean that if you approximate the earth as a solid ball of matter, as you get further down into the earths crust, gravity will "reduce" because the opposing mass at the other side of the earth will decrease the effective gravity.

However, this isn't the end of the story. The force applied upon the center of the earth is equal to the sum of the force exerted upon the earth by the entire mass still affected by gravity. You can argue that the ground isn't affected by gravity, and I can then empirically prove you wrong by throwing dirt at you. The sum of all the force applied by all the different layers of the earth, with all their different gravitational constants, is then multiplied by the area of the core. With that completed, you'll immediately see that there are massive pressures at the center of the earth. Those pressures increase the melting temperature of any substances unlucky enough to be that far down, resulting in a solid core surrounded by superheated magma. As an aside, the high energy at the core MAY result in a point gravitational field because the amount of energy would raise the relativistic mass.

Different gravitational constants? I'm not sure what you mean - g is always the same, seeing as it's...a constant.
There are massive pressures, and IMO Joe explained it the best - two masses half the size of Earth press upon the core, compressing it.
You have to take into account every force acting upon an object. If two objects are pulling on something equally, then the object is in equilibrium, and, all things being equal, it shouldn't experience any further acceleration. But it will be compressed! It's like two objects attached by springs, one at the top and the other at the bottom, and you are in between those objects. The springs will try to bring the objects together, but I would be in the way - I'd feel the force, accelerating me at the atomic level and causing pressure.