[dda]

Airplanes take off based on lift created by the passage of air over the wings. Air speed, not ground speed determines the point at which an airplane takes off.

Propeller planes gain airspeed by screwing through the air, pulling the plane forward. Jet engines gain airspeed by pushing the plane through the air. Aircraft carriers launch their planes through a combination of catapult and jet engine to get airspeed.

The ground is irrelevant except for the fact that the plane must take off from some surface.

The problem in the hypothetical is that the wheels and treadmill are irrelevant. A better question would be what if there was a giant wind machine behind the plane that increased exactly as the plane's thurst increased. If it did, the airplane could achieve any ground speed and still not be able to take off.

There is a reason why airports have runways facing in different directions and swithch the airplanes around so that, as far as practical, airplanes take off and land heading into the wind. Aircraft carriers always turn into the wind to launch or recover aircraft.

Also, if the head wind is sufficient, the plane could take off from a stationary start, it would seem. This is the principal upon which the wind tunnel is based.