OK, based on the original plane/conveyor statement, we find that the conveyor speed is dictated by the plane's speed. How is the plane's speed measured so that the conveyor can move at that speed, too?

1. The plane's speed is determined by a sensor on a wheel and the plane's speed is assumed to be based on the rotational speed of the wheel (like a car speedometer). As the forward thrust of the plane's engine(s) start moving the plane forward, the wheels start rotating.

At the first feedback cycle, the conveyor system detects a 1 mph rotational speed of the wheel, so the conveyor starts moving in the opposite direction at 1 mph. What's this? Now the wheel is rotating at 2 mph. OK, the conveyor does the same. Now the wheel appears to be moving at 4 mph, so the conveyor does the same.

Do the math. Very quickly, the conveyor speed will approach infinity. So then, the question becomes - based on the conveyor's measurement and feedback cycle, how fast will the plane's tires blow up due to rotating beyond their design limits.

2. The plane's speed is determined based upon measurements against a fixed point on the ground (not on the conveyor). When the plane is moving 10 mph relative to the fixed point, the conveyor also moves 10 mph in the opposite direction. The plane is moving through the air at 10 mph and the wheels are rotating as if it were traveling at 20 mph.

Let's say the plane can rotate and lift off the ground moving at a speed of 100 mph (air speed if no wind and also ground speed compared to the imaginary fixed point). The plane will lift off. Just before leaving the conveyor, the tires will be rotating twice as fast than if it were lifting off from a fixed runway.

In summary, the moving conveyor does nothing to affect the air moving across the plane's wings, only to affect the rotational speed of the plane's tires. The plane moves from thrust imparted upon the air by the propellors or turbines to move the body of the plane forward regardless of wheel speed and will eventually take off. (Disregarding, of course, the natural physical effects of wheels potentially spinning at some speed at which they were not designed to handle).

Edited to add: Now that this problem is solved, here's another one to consider: Using common mathematical equations, prove that 1 = -1.