Okay, now you take your feet off the pedals and put on a rocket backpack, which means you are pretty much the same as a plane (wheels for rolling, rocket thingy for propulsion). When you fire up the rocket backpack, it'll push your forward, right? But the treadmill will start going backwards, cause it tries to match your speed. But you're not pedalling anymore, so you won't care about what the bike beneath your does. And for that very reason, you will move forwards, such as a plane would on a conveyor belt.
I fully understand what are you trying to say, but you can't just say "you don't care" if you're doing a physical analysis. And
that is the whole point of this problem, it gives you different answers depending on what direction you look at it from. I'll chew it out while I'm in the mood
Continuing with your analogy - what will be the speed of the pedalbike wheels relative to the ground? It should be equal to your speed relative to the ground, since you're still sitting on the bike, and for the sake of accurate analogy, you can't leave the bike.
Wheel that is moving forward on the treamill MUST have higher speed than the treadmill itself if it has grip.
Treadmill matches this higher speed of the wheel.
But you have external force moving the wheel, making it even more faster than the treadmill.
But treadmill must follow the speed of the wheel by definition, so speed increases once again.
This continues to the infinity.
See the conflict? To move even a millimeter, speed of the wheel must be higher than speed of the treadmill, but speed of the treadmill must be equal to the speed of the wheel. You can't move because moving would... make you stationary.
You can even go as far as simplifying this problem to: can you move an object with wheels AT ALL if it sits on the treadmill bla bla bla... using ANY external force.
Physically this is impossible, because reaction can't possible happen at the precisely the same time as action, and of course you can't make wheels spin at infinite speed.
But as a thought experiment I'll stay at the "not flying" camp. Engine blasting off can move the air or the whole damn universe, but this treadmill is not going anywhere relatively to the plane bacause that's the condition of the problem.