Thomson's Lamp and the possibility of supertasks

Solution 1:

The following tries to avoid physical arguments:

The lamp $L$ has a well defined state $s\in\{0,1\}$ at time $t$ only if there is an $\epsilon>0$ (which may depend on $t$) such that $L$ is in state $s$ during the complete interval $\ ]t-\epsilon,t+\epsilon[\ $. It follows that the lamp is not in a well defined state at any of the switching times $t_n=2-2^{-n}$ $\>(n\geq0)$. Why then should we expect it to be in a well defined state at time $t_\infty=2\ $?

Solution 2:

The series $S = \sum \limits_{i=0}^\infty {(-1)^i}$ does not converge, so your calculation makes no sense to begin with (its Cesaro sum does converge to $0.5$, though).

Furthermore, a physical object such as lamp can't be turned off and on at arbitrarily short time intervals. As soon as you reach Planck time (approximately $5\cdot 10^{-44}$ seconds) really weird things are to be expected, though you'd probably be better off asking about it at physics.se. And that's completely ignoring the fact that it takes some time for the lamp to turn on (or off).

Solution 3:

The author is trying to conclude that an infinite number of tasks cannot be performed in a finite amount of time because there is a minimum amount of time required to perform any task. However, his argument is flawed. He has created a situation where it is impossible to deduce the state of the lamp and uses that to conclude that infinite tasks are impossible.

Consider a restatement of the problem. Suppose a green particle is moving at a constant speed and travels from point $A$ to point $B$. Halfway between $A$ and $B$ it changes color to red. Three quarters of the way it changes back to green, etc.

What color is the particle when it reaches point $B$?

I haven't told you. I said that it was green at point $A$ and I gave you a rule for determining its color at fractional distances from $A$ to $B$ but I have not defined what the color is at $B$.

Okay, so let's just say that at $B$ the color of the particle is whatever it was last. What color was that? It is undefined because for any time interval before the particle reaches $B$ there are an infinite number of points at which the particle changes color. It is equivalent to asking what the next rational number is after $1$ with the usual linear ordering. There is no next rational number and in this case there is no last color before reaching $B$.

Therefore, I have to tell you what color the particle is at $B$ the same way I have to tell you what state the lamp is in at two minutes.