I have measured out my life with adjustments
To my lightbulb contraption and the proliferation of its
-Thomas Stearns Edison
Everyone is of course familiar with famed inventor/poet T.S. Edison (the wizard of Faber and Faber), but what most people don’t know is that he spent the last years of his life building an elaborate light bulb contraption. The device consisted of a single bulb, which could emit blue, green, or red light, connected to a set of switches, each of which had three settings: blue, green, and red. Edison’s set of switches grew throughout his life (unconfirmed rumors suggest that, by the time of his death, the number of switches was greater than the size of the continuum), but, at any given time, there was a deterministic rule that associated each possible switch setting with a specific color for the bulb to emit. Edison tinkered with these rules but required that they all satisfy the following basic criteria:
1) If every switch is pointing to the same color, then that is the color of the bulb.
2) If every switch is changed, then the bulb must change color.
Let us call a rule that satisfies these two criteria an acceptable lightswitch rule. Just as we saw with elections last week, ultrafilters provide an easy way of generating acceptable lightswitch rules. Namely, let be the set of all switches, and let be an ultrafilter on . For any given configuration of the switches, there is a single color (red, say) such that the set of switches pointing to that color (i.e. red) is in . The rule associated with (call it the -rule) then says that the lightbulb should be red. This rule is easily seen to be an acceptable lightswitch rule.
T.S. Edison was aware of these ultrafilter rules but found them unsatisfactory and tried tirelessly to develop a different sort of acceptable lightswitch rule. Unfortunately, he never succeeded; in fact, we shall prove today that his search was in vain.
Theorem: Every acceptable lightswitch rule is in fact the -rule for some ultrafilter on .
Proof: Suppose that we are given an acceptable lightswitch rule and that is the set of all switches. We will define an ultrafilter on and show that our given rule is in fact the -rule.
First, a quick observation: if all of the switches point to one of two colors, then the lightbulb must be one of those two colors. Why? Well, suppose there were a situation in which all of the switches point to, say, blue or green, while the bulb is red. Now change all of the switches to red. By criterion (1), the bulb must be red. However, by criterion (2), since all of the switches were changed, the bulb must change, so it can no longer be red. Contradiction! We illustrate this, as we will illustrate our future arguments, with a picture. In our pictures, the set of all switches will be depicted by a rectangle. Imagine that the switches are physically located in the rectangle and that the color of a switch is determined by the color of the region in which it is located. The switch configurations are numbered in the order in which we imagine them occurring, and the pictures are labeled with the state of the lightbulb.
We are now ready to define our ultrafilter, . First, some notation. If , then we let denote the complement of , i.e. . Given a subset , we let if and only if when the switches are set so that all switches in are blue and all switches in are red, the lightbulb is blue. We must now show that is indeed an ultrafilter and that our rule is the -rule. We do this in a series of steps.
Step 1: and .
This follows precisely from criterion 1: if all switches in are blue, the bulb must be blue, while, if all switches in are red, the bulb is not blue.
Step 2: Suppose . Then either or .
Suppose that all switches in are set to blue and those in are set to red. By our prior observation, the bulb must be either blue or red. If the bulb is blue, then by the definition of . If the bulb is red, then change the switches in to red and those in to blue. Since all of the switches have changed, the bulb must change and therefore must be blue. In this case, , again by the definition of .
Step 3: Suppose , and are two different colors (chosen from blue, green, and red), and the switches are configured so that all switches in point to and all switches in point to . Then the bulb has color .
If is blue and is red, then this is precisely the definition of . We prove it here when is green and is red. The other cases are dealt with similarly. As before, we use a picture. In the picture below, the left side of the rectangle corresponds to and the right side to . We start with all switches in blue and all switches in red. Since , the bulb must be blue. We then perform a series of manipulations, and repeated applications of criterion 2 will lead to our conclusion.
Step 4: Suppose , , and . Then .
We again employ a picture. In the picture, the left half of the rectangle represents and the left three-quarters of the rectangle represents . We start with all switches in blue and those in red. We then change all of the switches in to green and those in to blue. We changed all of the switches, so the color of the bulb must change and therefore must be green. But then, by Step 3, we have .
Step 5: Suppose we are given a configuration of switches, possibly using all three colors, and is the set of switches matching the color of the bulb. Then .
For concreteness, suppose that the bulb is blue and is the set of all switches pointing to blue (so all switches in are pointing to green or red). Now change all switches in to red and all switches in to blue. The bulb must switch colors, so it must be red. But, in our new configuration, we are using only two colors, so, by steps 2 and 3, we must have .
Step 6: If , then .
Once more, a picture. We draw the picture in such a way that implies that . If this is not the case, then simply replace by . By step 4, this is in , and its intersection with is the same as . In the picture, the left of the rectangle corresponds to and the right corresponds to . Their intersection is thus the middle of the rectangle. Since , if all switches in are blue and all other switches are red, the bulb is blue (call this Configuration A). Since , if all switches in are green and all other switches are red, then the bulb is green (call this Configuration B). Now suppose that all switches in are red, all switches in are green, and all switches in are blue (call this Configuration C). Going from either Configuration A or B to Configuration C, all switches change, so the bulb must change. Therefore, the bulb in Configuration C can be neither blue nor green, so it must be red. By Step 5, this means that .
And now we are done! Why? Steps 1, 4, and 6 are precisely the verifications that is an filter, Step 2 shows it is an ultrafilter, and Step 5 implies that the given rule is the -rule.
P.S. I found this result in Komjath and Totik’s excellent book, Problems and Theorems in Classical Set Theory. Also, it is interesting to note that the result continues to hold (by essentially the same proof) for any finite number of colors greater than or equal to 3 but fails if only two colors are allowed. So here’s an exercise for you, reader: come up with an acceptable lightswitch rule for two colors that is not equal to an ultrafilter rule.