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Originally Posted by Sharur
I think it's already been stated that the plane takes off when talking about the treadmill going same speed as the plane and it being a paradox when the treadmill goes the same speed as the wheels of the plane.
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Actually, it's only paradoxical in the case when the plane is able to move. If the plane cannot move because of the action of the treadmill, then it is the wheels interpretation which makes sense and the "same speed as the plane" interpretation which is paradoxical.
Perhaps a better wording would be "The treadmill moves at the same speed as the plane would be going after the same period of having the engines on had the treadmill not been moving". That's entirely independent of the actual current speed of the plane, and therefore works under both answers.