OK, I read the post on Jason Kottke's site about the The case of the plane and conveyor belt, and quickly concluded that the plane cannot take off.
I was shocked, SHOCKED, to find smart people like Matt Haughey and Mike Buffington believing that the plane can take off.
Let me clear this up.
For an aircraft of any type to fly, it must overcome the constant acceration downward of gravity. Flying, for the most part, has little to do with speed and everything to do with the lifting force that counteracts the pull of gravity. Got that? Lifting force is all the matter to leave the ground and fly.
The question then is how do you generate enough lifting force. With a balloon, the lifting force is based on the fact that a bag of lighter-than-normal-air gas (usually hydrogen, helium, or hot air) will rise above normal (cold) air. Make sense? A balloon has no direct thrust or "push against air", it simply has the lifitng force of the gas.
Now let's look at an airplane. How does an airplane fly? An airplane flys off the ground solely because it can generate a lifting force. An airplane generates this lifting force by air flowing over an airfoil (known as a wing). The faster the air travels over the wing, the more lift is generated.
Planes that go fast can generate a lot of speed (air moving past the wings) can generate a lot of lift. If an airplne goes too slow, it doesn't generate enough lift to counteract the force of gravity, and it falls to the ground. This is called a stall.
So, now that you are versed in the basics of aerodynamics, let's look at the question of the airplane on the conveyer belt. The airplane will sit on the conveyer belt with engines blasting, tires rolling, and to the outside observer, standing in place. While tremendous forward force is being generated, there is no increase on the airflow over the wings.
So, if there is is no increased airflow over the wing, there is no lift. Without lift, the airplane CANNOT take off and fly. Let me repeat that key part, without lift, the airplane CANNOT take off and fly. It doesn't matter how hard the engines 'push the air', there is no lift being generated in the conveyor belt scenario.
I hope you can now see clearly the correct answer to the 'question' is that the plane will not take off.
If you start to argue about the validity of the basic aerodynamic theory of lift, you might as well start arguing that the earth is flat or in creationism.
Update: After reading the comments it seems like the debate is hinging on whether the plane stays stationary or not. Obviously above I'm assuming the plane/conveyor belt contrapion is designed to hold the plane still. If that is not assumed then then the entire question is stupid and ridiculous. It's then like asking can a plane take off in the rain or snow when the wheels slip.
If that's really the point that the question poser is making then it's fucking stupid. It's like those stupid tests where the first line says "1) Read all instructions carefully before taking the test" and the last line says "57) Now that you've read all the instructions carefully, don't do 2-56, write your name in and turn in the test." Fuck that shit.