Christopher Campbell wrote:

The other gotcha in this little puzzle is that it attempts to get you to
divide by zero.

Explain to us please where the statement of this problem ever involves
division by zero.
One can readily see where the statement implies a value of zero for air
speed since in the absence of wheel slip:

Treadmill speed = wheel speed (stated explicitly in the problem)
and
Air speed = wheel speed - treadmill speed (assuming calm air)
this directly implies that
Air speed = 0.

But I don't see where division by zero ever comes into play.
The stated problem does imply a runaway positive feedback in the
treadmill speed control. I.e. the moment the plain starts to roll
forward the control system would speed up the treadmill to match the
wheel speed. The motion of the treadmill would then speed up the wheel
rotation to a higher speed thus forcing the treadmill to move still
faster to catch up. The result would be an ever increasing treadmill
and wheel speed until something gives - most likely the tires (if we
ignore the technical difficulty of building the specified treadmill).

This is the old Achilles vs. the Tortoise conundrum that so
puzzled ancient Greek mathematicians. The puzzle was this: Achilles and a
Tortoise agree to have a race. Achilles agrees to let the Tortoise have a
head start of getting half way to the finish line. The starting gun sounds
and they are off! (Well, the Tortoise is, anyway.) The Tortoise reaches the
half-way mark and Achilles starts running. But by the time that Achilles
reaches the half-way mark, the Tortoise has moved forward. And by the time
that Achilles reaches the point where the Tortoise has moved to, the Tortoise
has moved forward again, albeit not as far as before. Again Achilles reaches
the third point where the Tortoise was, but the Tortoise has moved forward
again. No matter how fast Achilles runs, he can never catch up with the
Tortoise. It was this sort of logic that led the Greeks to conclude that
everything was imaginary and that motion was impossible. They could not solve
the problem because they did not have the number zero.

Zeno's Paradox. But I doubt if you could find any ancient Greeks who
actually concluded that motion was impossible. Even while puzzling
with Zeno over his problem, they continued to go to the markets to do
their shopping and to their respective work places.

And there's no need to have the concept of the number zero to solve
Zeno's paradox, just the idea of the convergence of some types of
infinite sums. I.e. if each successive run of Achilles is half as long
as the previous one (say he walks twice as fast as the tortoise) then
we have a sum for the total distance 'D' of the form:
D = x + x/2 + x/4 + x/8 +...
multiplying this by 2 gives:
2D = 2x + x + x/2 + x/4 + x/8 + ... = 2x + D
subtract D from both sides and we solve for the total distance Achilles
needs to walk:
D = 2x; i.e. twice the distance of the headstart he gives the tortoise.

The airplane-on-a-treadmill is just a restatement of the same problem. It
attempts to convince you that the airplane cannot move relative to an outside
observer if the treadmill always moves at the same speed as the wheels. If
the wheels accelerate, then the treadmill accelerates, so the plane cannot
move, right? Wrong. The airplane does move, and it accelerates relative to an
outside observer at the same rate as it would if the treadmill remained
stationary. The only thing that changes is that the wheels spin faster.

Sure, but airplane wheels have some maximum speed. Once the treadmill
gets up to that maximum speed the airplane wheels would fail and the
airplane is now sitting on a treadmill with a bunch of failed tires.
So the question becomes whether a plane can still take off after you
shoot out all the tires when it first begins its takeoff roll.

And yes, postulating a frictionless surface for the treadmill gets
around the problem and allows a normal takeoff. But the very term
treadmill implies a surface with reasonable friction, i.e. the tread.