Well actually it is true (It is known as the "shell theorem" which I was not aware of when I wrote my post) and you can find a proof in any elementary book on classical mechanics, c.f. wikipedia. I will be happy to help you with the required integration if neccesary:
It's still not true. Please do show the math. I would suggest you pick the exact center of the sphere and, for a good distribution of test points, five random places inside the hollow sphere and run the math. Oh, and please be sure to use relativity when describing the gravity. Classical mechanics is so hamfisted at it.
Now, so returning to the mineshaft, if we assume the earth is a body of uniform density the gravitational pull at a radius r will (per 1 of the shell theorem) scale as the mass (proportional to r^3) divided by square of distance (newtons low of gravitation) and so scale as r.
So, even using classical mechanics, it's not, as you said, "no acceleration in a hollow sphere". Yeah, I did some gravitational math earlier on the thread. It's not very complicated.
Ofcourse if you make assumptions on the density of earth this may affect the result, however my comment was discussing the idealized situation and at any rate I wont look up the density of the mantle compared with the inner parts of the earth now.
Yeah, I did that the other day. It's a tougher number to find tham I thought it would be, lots of assumptions have to be made.
Anyway, at the exact center of a perfect sphere of perfectly uniform density in a non-moving system (that bit's important) with no other external forces whatsoever, there isn't "no gravity". There is equal acceleration in all directions. Any particle or body placed in and only in the exact center of a sphere under such conditions would experience the exact same gravitational acceleration in all directions and be held in place.
No gravity would mean it could float around and experience zero gravitational forces at all in any direction.
Hope that clears it up! Looking forward to seeing your math!