That Plane Treadmill Thing

There's a common internet debate that asks if a plane on a treadmill, where the treadmill moves backwards as the plane moves forwards (thereby canceling its movement), if the plane will successfully take off. Apparently, depending on how the treadmill's actions are conceived, the answer is "yes" or "Eventually."
If the treadmill matches the wheel's speeds, then the push from the engines will accelerate the plane to a speed such that it takes off, relatively quickly. Not as fast as a plane on a cement or asphalt runway, but still relatively quickly.
If, however, you insist that the conveyor belt somehow also matches the engine speed plus the wheel speed, you end up with a paradox where soon neither one can be a real number. Either the conveyor belt jams at some point, at which time the plane takes off instantly, or the wheels tear from the plane due to friction (and the now unhindered plane takes off, albeit in a state that will require a crash landing). In fact, that answer amounts to mathematical nonsense.
Why? Let us say that the plane's wheels are Wb, the conveyor belt is Wc, the plane's engines produce We. The plane's velocity, Wv, is defined by: Wv = Wb + We - Wc. If Wc = Wv, then Wb + We must equal zero. Otherwise, as Cecil Adams puts it, "A + 5 = A." Since the plane is attempting to take off, We is probably greater than zero. Therefore, Wv != Wc.
QED.
And even if you reject that, the plane's velocity will reach Aleph-1, the conveyor belt will reach Aleph-0. Aleph-1 is greater, the plane takes off. Also QED.