infinity paradoxes

Hotel Infinity

Suppose that, maybe somewhere in New Jersey, there is a hotel with an infinite number of rooms. You arrive late one night and ask the front desk clerk if they have a vacancy. He replies that every room is occupied, however, he can arrange for you to get one. But how, you wonder, if there is no vacancy? The answer is simple: the clerk will simply ask the people in room 1 to move to room 2, those in room 2 to move to room 3, those in 3 to move to room 4, and so on. Since there is an infinite number of rooms, everyone will have a room to move into, and room 1 will be available for you.

Hotel Infinity is an amazing place, you think to yourself as you sign in. But just as the clerk is about to give you your key, an infinite number of people arrive for an APA convention. The clerk cleverly figured out how to get you a room, but can he accommodate an additional infinity of guests? Amazingly, he can. He just asks everyone to move again, but this time to the room number that is twice the number of their current room. In other words, you would move to room 2, the people in 2 would move to 4, those in 3 to 6, those in 4 to 8, and so on. This will leave all odd numbered rooms — an infinite number of them — vacant.

This paradox illustrates an unusual property of infinite sets. With finite sets, a (proper) subset will always contain fewer members than the entire set. A part is smaller than the whole. But with infinite sets that is not the case: one part of the set can be just as large as the whole. For example, there are as many even numbers as there are natural numbers, even though the natural numbers contain all the even numbers plus the odd ones as well. This can be seen by pairing the natural numbers with the even numbers to show that there is a one-to-one correspondence between the two sets: 

 
1   2   3   4   5   6  ...
 |    |    |    |    |    |
2   4   6   8  10  12 ...

Likewise, even though only some numbers are perfect squares (1, 4, 9, 16, 25, ...), and the distance between each perfect square becomes greater and greater as we progress down the number line, there are as many perfect squares as there are natural numbers. For each natural number is the square root of exactly one perfect square. (This is sometimes known as Galileo's Paradox, as it was first pointed out by the famous Italian physicist and astronomer.)

A variation on hotel infinity results in an interesting Zeno-style paradox. Contrary to what might first be supposed, the hotel doesn't have to occupy an infinite space. Suppose the hotel has one room per story. If each room is half the height of the one below, then the entire structure will be only as tall as a two-story building. But if that's the case, then it should have a roof on top. And if it has a roof, then, as any reputable architect can point out, the other side of it ought to be the ceiling over some room. However, what room will that be, given that the hotel has an infinite number of them and therefore no top story?


The Infinite Circle

Nicholas of Cusa (1401-1464) made the following interesting point regarding the shape of an infinite circle. The curvature of a circle's circumference decreases as the size of the circle increases. For example, the curvature of the earth's surface is so negligible that it appears flat. The limit of decrease in curvature is a straight line.

An infinite circle is therefore... a straight line!


The Paradox of Tristram Shandy

This paradox, formulated by Bertrand Russell, is based on the 18th century novel The Life and Opinions of Tristram Shandy, Gentleman, by Laurence Sterne. Here is Russell's statement of the paradox:

"Tristram Shandy, as we know, took two years writing the history of the first two days of his life, and lamented that, at this rate, material would accumulate faster than he could deal with it, so that he could never come to an end. Now I maintain that, if he had lived for ever, and not wearied of his task, then, even if his life had continued as eventfully as it began, no part of his biography would have remained unwritten."

Suppose Tristram Shandy continued at the painfully slow rate at which he started, so that he took a full year to write about each day of his life. In spite of this, there is a one-to-one correspondence between each year that he writes in and each day he writes about. Therefore, no matter what day of his life you care to consider, there will eventually come a year in which he will be able to write about it. There is no part of his life that can never be written down. Nevertheless, he gets further and further behind!

It's interesting to reverse this paradox and consider what would happen if Tristram had already been writing for an infinite amount of time. It at first may seem that the two temporal directions might be mirror images of one another. In that case, just as he might begin to write at time t in the original paradox, in the reversed version it would seem he might have finished his task at t. But a little reflection shows that that is impossible. If he had just finished writing his autobiography, then he would have just written about the most recent day of his life. But since it takes him a year to write about each day, he would have had to start writing about this most recent day 364 days before the day started! Thus, unless Tristram can foretell the future, he cannot have finished writing yet, even though he has already spent an infinite amount of time on the task.

Suppose Tristram has in fact been writing forever and has just finished describing another day. When might have been the day he just finished writing about? As we've just seen, he could not have been writing about today, for he would have had to start writing about it a year ago. So it seems that the most recent day he could have been writing about is a year ago today. But then what was he writing about in the previous year? He would have been writing about a year ago yesterday. But that too is impossible, for he would once again have had to start 364 days too soon. Repeated application of this argument shows that, no matter what date in the past one chooses, Tristram could not already have written about it. He therefore can only have finished writing about a day that lies in the infinitely remote past!




©2000 Franz Kiekeben