That is the point of putting a plane on the treadmill, to keep it stationary - to keep it from having forward motion. I didn't assume anything, that's just physics in motion (or lack of motion

). I've seen/heard this 'teaser' before.
So what you are saying is essentially, the plane will have forward motion regardless of what the wheels are on? The point of having the treadmill is to keep up speed on the wheels so there is no forward motion. In actuality, there is NO motion relative to air/ground. Only motion relative to the treadmill, which (is supposed to) keeps the plan at a standstill. And again, we are talking about theory, which requires a perfect model of physiscs (no friction). This also is taking into consideration that the treadmill is programmed to keep up to speed with the plane, countering any forward motion, which would lead to the eventual destruction of the treadmill, and that inturn would let the plane have forward motion.
Now, if the treadmill is NOT a powered or programmed treadmill, then yes - the prop/enginge will produce thrust as normal and the plane will eventually overcome the counterforce of the treadmill (if there is any) and will take off as normal.
Two different interpretations of this teaser. That's one thing people usually don't mention when they ask you this - is the treadmill powered or not?