747 on a treadmill conundrum..

[SteelCurtain]

All of this is moot, though. The plane's wheels have effectively zero effect on the plane's ability to create thrust and move forward on the runway. If the treadmill is always set to match the forward velocity of the aircraft, then the 747's wheels will always be rotating at 2x the indicated airspeed. The plane will take off just fine.

So we could make really short aircraft carriers?

We do make really short aircraft carriers. A normal runway is 1.5 miles long. An aircraft carrier is 1100 feet long. And the takeoff section of the aircraft carrier is only 300 feet long. The plane is hooked to a steam powered catapult and hurled into the air from zero to about 150 knots in less than 2 seconds.

The point still stands. The plane's wheels are not how the aircraft translates thrust into airspeed and therefore lift. The engines push massive amounts of air backwards, which has the equal and opposite effect of pushing the plane forwards. It works the same way as a helicopter. If a helicopter with wheels as landing gear were on a treadmill moving at any speed, the treadmill would have negligible effect on the helicopter's ability to generate thrust, and therefore take off.

It is the propeller/engine/turbofan that makes the thrust by pushing against the air, not by driving the wheels.