If the plane moves at a given speed in one direction, and the treadmill moves at that same speed in the opposite direction, what is the net speed of the plane? The debate is not: "how does the treadmill know what speed the plane is going?" it simply says the speed is matched, thus cancelled.

You're assuming that, in rotating, the treadmill exerts a backward force on the body of the plane. It's only interacting with the wheels.

Flint, when the treadmill moves, what part of the plane is it moving?

The wheels.

The wheels are free-spinning around their axles.

So, as the plane moves forward (which i thought you said it couldn't do ;)) the wheels end up spinning faster and faster as the treadmill matches the forward motion of the plane.

If the plane is moving forward at X mph, and the treadmill is moving backwards at X mph, then the net forward speed of the plane is X mph, but the wheels are spinning at 2X mph. The treadmill is matching the speed of the plane, not the speed that the wheels are turning.

another way to look at this is to consider the aircraft carrier: the planes are 'launched' with a catapult. But the wheels are on the deck, and the catapult moves much faster than the deck. The deck could be moving backwards and still the catapult could launch the plane.

Did my bike example make sense to you?

In order for the question not to contradict itself, you have to assume that the plane "moving forward" is only relative to the treadmill. It cannot "begin" to move forward, in any other sense, if the treadmill is moving backward at the same speed, as stated. What the wheels are doing doesn't matter. How the treadmill works doesn't matter. The stated scenario is that the plane cannot ever move forward because the treadmill moves backward at the same speed - not the same speed as the wheels, the same speed (realtive to the treadmill) as the whole plane.

No, I'm taking the question literally and establishing the parameters of discussion, to prevent going on irrelevant tangents.

Yes it makes perfect sense to me, as a description of a bike, a person, and a treadmill. We don't need "another" way. We need to read the question and not add anything to it.

Right. The treadmill is preventing the plane from moving forward because it matches it's forward speed.

Why does the treadmill moving backward prevent the plane from moving forward?

Because it matches its forward speed, as stated. X minus X is zero.

Nope. That is one interpretation, but that interpretation requires you to assume that planes don't work the way they do. It is not possible for the speed of a treadmill to affect the speed of the plane (short of mechanical failure in the wheels). Therefore, your interpretation is incorrect. You must assume that the plane is moving forward relative to the ground, and the treadmill is moving backward at the same speed, relative to the ground. That is the only physically possible interpretation.
Nope, the plane keeps going forward, the treadmill moves backward, and the wheels do double duty.

It doesn't have to, and I never said it did. Where did you get that? If the treadmill matches the forward speed of the plane, the plane cannot "begin" to move forward, except relative to the treadmill.

But the treadmill X can't subtract from the plane's X. The plane still goes X, but the wheels go 2X.

It can, because this question says it can. Start by establishing what is being discussed.

When you said that they could cancel each other out. That is not physically possible.The treadmill is moving backward relative to the ground at speed X. The plane is moving forward, relative to the ground, at speed X. The plane is moving at speed 2X relative to the treadmill. Your interpretation is not physically possible.