It seems that necessarily:
- If an object was F at t1 and non-F at t2, and the object existed at both of these times and all intermediate times, then the object changed with respect to F.
But suppose that a lamp is on at all times that are rational numbers and is off at all times that are irrational numbers. Then according to (1), the lamp changed with respect to being on. Yet when did it change? A change is always a transition from one state to another. Suppose I take an ordinary lamp that's turned on at noon, off at 1, on at 2, and off again at 3. Then we can say that it changed from off before noon to on at noon; from on before 1 to off at 1, and so on. But there was no transition from on at noon to off at 3--there was, instead, a series of transitions. Likewise, then, if the lamp is on at t1 (which then is rational) and off at t2 (which is rational), it didn't transition from on at t1 to off at at t2. For after t1 and yet before t2, then lamp was off and then on and then off and so on, infinitely many times. So just as the ordinary lamp didn't transition from off at noon to on at 3, the weird lamp didn't transition from on at t1 to off at t2. Hence, it never transitioned.
What should we say? I guess we have three options. First, deny the possibility of a dense time sequence. Second, deny (1). Third, deny that change requires transitions.