A lamp is switched on and off at the end of each half of the remaining minute. The times add to a finite limit, so every switch happens before the minute is up. The state after each switch alternates, so it has no limit. The series settles when the switching ends and leaves the lamp's state at that moment unfixed, because there is no last switch to fix it.
Core ideas
The two earlier lessons showed that infinitely many tasks can fit in a finite time. This one asks what the world looks like when they are done.
Each switch of the lamp is a task. They come faster and faster, so that all of them, infinitely many, are over by the end of the minute. A series of times that has a finite limit is what makes the story describable.
The time of the nth switch is 1 − 1/2^n, which approaches 1. The state after the nth switch is on if n is odd and off if n is even, and the sequence on, off, on, off, ... approaches no value.
Every switch has a successor. A rule that says what the lamp does after each switch says nothing about the lamp at the end of the minute, since that is after all the switches and not after any one of them.
The material
The story, the two sequences, and the question the story does not answer.
A lamp starts off. After half a minute it is switched on. After a further quarter of a minute, three quarters in all, it is switched off. After a further eighth it is switched on, and so on, each switch coming after half the time that remains. The minute has infinitely many switches in it. Thomson's question is: at the end of the minute is the lamp on or off?
The argument ends where no one can stay: a lamp that is neither on nor off. Thomson drew the conclusion that the setup is impossible, which is to deny the second premise. It is worth seeing what the setup does and does not fix before deciding that.
The lab's lamp setting shows both sequences. The switches happen at the partial sums of the halves, so the nth switch is at time 1 − 1/2^n of the minute. After ten switches the lab reports the time 1023/1024, the same figure as in “Zeno's Dichotomy”. Those times converge to 1.
The second sequence is not a number that approaches anything. The term tile shows the state after n switches: with ten switches it reads off, and with eleven on. Odd gives on and even gives off, because the lamp began off and each switch reverses it. The terms never settle toward either, so there is no limit state in the sense that there was a limit time. The second preset is eleven switches: the state flips while the time moves only from 1023/1024 to 2047/2048.
Now look at premise four. The setup supplies a state for every moment before the end of the minute: at each time 1 − 1/2^n there is a definite state. It supplies none for the end of the minute, because that moment is not the time of any switch, and the rule says only what the lamp does at each switch. A rule about how the state changes at each step, applied to every step, says nothing about what follows all of them. “Every time it was switched on it was later switched off” is true of every moment before the minute and silent about the minute itself, and the fourth premise reads the rule as if it fixed a state there.
That is one way to reject Thomson's conclusion, and it is the standard one: exit one, against the fourth premise. The story is consistent, because its description of the lamp before the minute is complete and consistent and does not mention the state at the minute. Whatever the lamp is doing then, it does not violate the description. A rule that did say, such as “the lamp is on at the end of the minute”, would be a further stipulation by the one who tells the story, and either choice is consistent with the rest. The price is that a story can describe a lamp at every moment of a minute but one and owe nothing about that one.
The case against this reply has force too. A physical lamp has a state at every moment, and if the setup describes a possible lamp then the end of the minute has a state that the setup does not say. A reader who finds that unacceptable has concluded, with Thomson, that the story describes nothing possible. That is exit one as well, against the second premise, and its price is saying which law forbids a body to be switched in this way. The two replies deny different premises and pay different prices, and the lesson's claim is the narrower one the lab can check: the series fixes the time and not the state.
Interactive
n terms added as fractions, against the limit when there is one
Method
Four moves, in this order, for any story with infinitely many steps in a finite time.
Find when the nth step happens, as a fraction, and its limit. If the limit is finite, every step falls before it.
Record the state after each step. Decide whether it approaches a value. An alternating sequence approaches none.
A last step would fix the state after it. If every step has a successor, the rule that moves from step to step does not reach the limit time.
The time is settled by the series. The state is settled only if the story adds a rule for it, and a lab cannot add one.
Both replies are exit one. The standard reply denies that the rule fixes a state at the end of the minute, at the price of a story that is silent about one moment. Thomson denies that the switching is possible, at the price of saying what forbids it.
The lamp starts off. Switch n comes at 1 − 1/2^n of the minute, and each switch reverses the lamp.
After ten switches the lamp is off and 1023/1024 of the minute has passed; after eleven it is on. The series fixes the minute and leaves the state at the end undefined, since no switch is the last one.
Given a supertask, you should be able to compute the time of the nth step and its limit, write the state after the nth step, say whether the states converge, state which question the story answers and which it leaves undefined, and name the premise each reply denies.
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