But the airplane has a verifiable answer.

That’s just the thing though. Everyone claims to have the objectively correct answer. But there’s no broad agreement as to what it is.

If the starting force is applied to the wheels, which push off the ground, then the airplane will remain stationary.

Is that really even an “airplane,” though? If that’s true of the hypothetical in the question, I’ve never heard it mentioned, and it seems counter to the spirit of the question.

But if it uses jet engines or props, which push off the air, then the plane will move forward.

But the formulation I’ve heard of it says that “the treadmill goes at exactly the same speed as the wheels rotate.” It follows that the plane cannot be moving either forward or backwards. (Assuming no slipping of the wheel against the treadmill, which I’ve heard explicitly stated in the question, If the plane is moving forward, then the wheels are necessarily rotating faster than the treadmill. If backwards, slower.)

The versions of the question I’ve heard specify there’s no wind, so the only way I can see for that to be true is if the friction of the wheel against the axle produces sufficient backwards force on the airplane to keep it from moving forward.

But the versions of the question I’ve heard also specify no friction between the wheel and axle. So either there’s something putting extra backwards force (I don’t want to call it “drag” because I believe that term’s usually reserved for air resistence) on the plane, which I definitely don’t think is in the spirit of the question or the question itself is nonsensical.

So, the proper answer is neither “the plane takes off” nor “the plane does not take off.” It’s “either the question itself is self-contradictory and thus unanswerable or it leaves out important details, in which case if I’m allowed to make up more details, I can make the answer whatever I want.” (If I decide the plane is tethered to something to keep it from moving forward, the answer is almost definitely “it doesn’t take off”, unless the propellers themselves produce enough wind to create lift. If I instead decide the treadmill itself is moving “forward” (from the plane’s perspective) relative to the air, then the plane does achieve lift.)

Or it’s possible you’ve heard a different version of the question than I have that, for instance, didn’t specify that the treadmill goes exactly the same speed as the wheels. If that’s the case, then yes, all else being equal, the plane would take off. If it instead didn’t state no friction between the wheel and the axle, I think the most reasonable answer, though one still not in the spirit of the question, would be that the plane would not take off.

One more thing to mention. I know MythBusters tested this one and that the plane took off, but the wheels of the plane and the surface of the treadmill either didn’t go the same speed or the wheels slipped against the treadmil. (As evidenced by the fact that the plane moved forward.)