Here are some follow-up questions to class, and to the “parable” I posted last time. (They are certainly not the only questions one could ask—I encourage you to think of more!)
In talking about this, it will help to use the term “countable”. A set is “countable” if it can be put into one-to-one correspondence with the natural numbers, {0,1,2,3,…}. In class, I showed that the even numbers are countable, and that the rational numbers (all fractions a/b, where a and b are integers, b not zero) are countable. I also showed that the real numbers in the interval [0,1] are not countable; that is, the real numbers are a “larger” infinity than the rationals or the naturals. You can take those facts for granted in what follows.
Question 1: Let’s say that god starts with a countable number of angels. (Since they are countable, god might as well give them identification numbers: 0, 1, 2, 3, …) This question is about how much room god needs for the angels. The rule for an angel is that each angel can take up as little volume as god chooses; but an angel cannot take up zero volume.
(a) Explain how god can fit any finite number of angels into a finite volume.
(b) Explain how god can fit ALL the angels into a finite volume. (Hint: Look back at our discussion of calculus, and Zeno’s infinite series.)
(c) Now, suppose god is building clubhouses for all the angel clubs, potential or actual. The rules for a clubhouse are the same as the rules for angels: god can make them as small a volume as she likes, but she cannot make them zero volume. Explain why she cannot fit all the clubhouses into a finite volume. (Hint: This one is a little tricky. Suppose god has a finite volume V to work with, and suppose the clubhouse volumes are all chosen. List all the clubhouses of volume bigger than (1/2)V—how many can there be? Then list all the clubhouses of volume bigger than (1/3)V that you haven’t already listed—how many can there be? By continuing this way, show that the number of clubhouses that fit into the finite volume must be countable. Because of this, conclude that not all clubhouses can fit into the finite volume.
Following up on the previous question: suppose that god had infinite space to work with. Let’s take “infinite space” to mean ordinary R^3 coordinate space, all points with x, y, z coordinates, where x, y, and z can be as large as we like. Even in infinite space, the number of clubhouses that will fit are still at most countable. For this reason, the clubhouses will not all fit, even in infinite space.
Question 2 (optional): Here is the outline of the argument why the number of clubhouses in infinite space is still countable.
(a) Explain why infinite space R^3 is made up of a countable number of pieces of finite volume. (You could use cubical blocks and a spiral counting argument like I did in class. More simply, you could imagine concentric spherical shells.) In each of those countably many pieces, only countably many clubhouses will fit, according to the previous question.
(b) Explain why, if we take a countable collection of volumes, each containing countably many clubhouses, we still get only countably many clubhouses in general. (Hint: Since the collection of volumes is countable, we can label them with natural numbers, say V_1, V_2, V_3, … Now, within each volume, the collection of clubhouses is countable, so we can also label them with natural numbers. The clubhouses in V_1 we can label C_11, C_12, C_13, … The clubhouses in V_2 we can label C_21, C_22, C_23, …. Now, there are different ways you can show this collection of all C_ij is countable. One way is to think of each clubhouse as labeled by a pair (i,j), draw all the pairs on an x-y plane, and make a zig-zag argument like I did in class.)
The arguments above lead to a very weird thing that is difficult to imagine: let’s consider a 1 x 1 square in R^2, the ordinary x-y plane with real coordinates. In class, I proved that the rational numbers are countable. By the same argument as the last part of the previous question, the set of all points in the square with rational coordinates (x and y both rational) is also countable. Because there are infinitely many rational numbers between any two numbers, if you imagine the set of all points with rational coordinates, it would visually fill up the whole square. (Right?) Mathematically, there would be points missing, but for any missing point, there are rational points which approximate it as closely as you like (in particular, more closely than the resolution of any actual picture).
Suppose that we surround each and every rational point by a small square. And suppose that we add up the area of all those squares. How much area do we get? Any guesses?
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… Certainly it seems like the total area should be at least 1, right? Because the squares would effectively have to cover over the entire 1 x 1 square?
No! In fact, I can find such squares with area less than 1/1000.
Here’s how: since the rational points are countable, I can label them in order: p_1, p_2, p_3, p_4, … Each p_i is a point in this square, with rational coordinates; and every point in the square with rational coordinates appears somewhere in this list.
Put a little square box around point p_1, and make the box small enough so it has area 1/2000.
Put a little square box around point p_2, whose area is 1/4000.
Put a little square box around point p_3 of area 1/8000.
The total area of the boxes is
1/2000 + 1/4000 + 1/8000 + 1/16000 + 1/32000 + …
What does that area add up to?
???!!!
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