The plot involves a science fiction scenario in which the human military is battling aliens in "tau space". Whenever we send a single ship into tau space it becomes replicated into infinitely many copies which can be manipulated independently. However, the enemy we are fighting also has infinitely many ships in tau space, and their ships are at a higher density than the human ships and so we are losing.

The story then introduces the obnoxious and annoying character of Dr. Binding Phipps. Phipps is a young mathematician, engaged to a senator's daughter. The senator does not like Phipps and hopes that his attempt to use mathematics to solve the problem fails...he is not disappointed. Phipps claims that "Cantor's Theorem" proves that there are the same number of alien ships and human ships and that this can be used to our advantage. Unfortunately, this does not work, apparently because Cantor was wrong. The senator (and the author of the story) suggest that mathematicians frequently believe falsehoods like Cantor's Theorem because we are unwilling to acknowledge that great mathematicians can make mistakes.

I have no reason to think it is true that mathematicians are inclined to ignore errors made by our "stars". That suggestion alone seems to me to be slanderous, though it would be interesting to see whether Anvil has any evidence to back up his claims. (My impression, after years in academia and mathematics, is that there is no other area of human activity in which the practitioners are as skeptical or honest about errors as mathematics.) However, to take a real mathematician such as Georg Cantor and to unfairly use him as an example of someone who has made such an error is extremely offensive.

I think it is probably the case that Anvil really thinks that Cantor made a mistake. Perhaps he even tried to explain this to mathematicians and interpreted their lack of support as blind faith of Cantor's work simply because he is famous. However, I assure you that I would be perfectly willing to acknowledge Cantor's error...if he had made one. In fact, I think it is simply a misunderstanding on the part of Anvil.

Let's first talk about what Cantor actually did and then from there go on to talk about Anvil's story.

Georg Cantor created a branch of mathematics that is able to compare the "size" of any two sets, even sets of infiinite size. This is interesting because prior to his work, one could compare two finite sets and say whether they had the same size, but for infinite sets all one could do was say "Well, they are both infinite!" Cantor made a definition that two sets have the same size if it is possible to pair up the elements of each set so that there is a one-to-one correspondence between all of the elements in one set and all of the elements in the other. For example, according to this definition, the set of even numbers and the set of odd numbers have the same size because we can pair up 1 with 2, 3 with 4, and more generally pair up the odd number x with the even number x+1. Since each odd number is paired with exactly one even number, and each even number paired with exactly one odd number according to this pairing, the two sets have the same size. That's not so surprising. A little bit more surprising is the fact that the set of even numbers and the set of all whole numbers also have the same size according to this definition! That seems strange because one might think that there are half as many even numbers as even and odd numbers together. (In fact, one could come up with alternative definitions of size which would reflect this.) But, since we can pair up the whole number x with the even number 2x, according to Cantor's definition the two sets do have the same size. More impressively, one can show that the set of all fractions too can be paired with the set of whole numbers so that the integers and the rational numbers also are infinite sets of the same size. But even that is not the interesting part of Cantor's work. The interesting thing is that not all infinite sets have the same size when using Cantor's definition! In particular, the set of all real numbers is actually bigger than the set of integers. More generally, what is known as "Cantor's Theorem" is that the set of subsets of a set is always a bigger set than the set you started with, even if the set you start with is infinite in size.

Now, back to Anvil: Anvil gets this wrong in several ways. For one thing, he claims that it is Cantor's Theorem which says that the number of alien ships and the number of human ships are the same. Quite to the contrary, Cantor's Theorem opens up the possibility that even though they are both infinite sets, they may not be the same size! The fact that the story never mentions this possibility suggests to me that Anvil may not have grasped this very important aspect of Cantor's work. Whether the two sets have the same "size" or not is a matter of definition...you can accept the definition or not. But, in the story the mathematician claims that it is necessarily possible to pair up the human ships and alien ships because both sets are infinite, and that is not true. Cantor's Diagonalization Proof shows quite clearly that you can have two infinite sets and that there would be no way to ever pair them up.

Moreover, Anvil's idea of where Cantor went wrong is that you can only pair up a finite number of elements of the two sets at a time, leaving an infinite number unpaired. If one had to physically move the elements of the set around, then such a physical restriction would be a problem. (In particular, it would be a problem in the case of the human and alien spaceships if one really could move only a finite number of them at a time.) So, I agree that Cantor's work would not be able to solve the problem in the story. (Phipps was silly for thinking that it would.) But, that is not a problem with Cantor's work, only a problem with the idea of applying it here. If I had infinitely many boxes, labeled 0, 1, 2, 3.... and there was a ball only in the even boxes, then I would not be able to physically move the balls to fill all of the boxes. But, this does not contradict Cantor! Let me explain in two ways (1) Suppose I had infinitely many empty boxes labeled 0, 1, 2, 3,...and I also had infinitely many balls labeled 0, 1, 2, 3.... so that there was no doubt that there were the same number of each. I still would not be able to ever fill all of the boxes. This is a physical restriction, but not an indication that there is an error in Cantor's claim that the set of boxes and the set of balls have the same cardinality. (2) To set up a 1-1 correspondence between the balls and the boxes does not require physically moving them. I can instantaneously make the rule that the box x is associated with the ball in box 2x. This is a 1-1 pairing. It is as good as taking a bag of apples and a bag of oranges and showing that there are the same number of each by pairing them off.

This story was originally published in If magazine in 1974 and, although I would have strongly argued against its publication on the grounds that it is confused and offensive, it was reprinted in the collection Interstellar Patrol II.

I received a copy of this story in an anonymous, manila envelope...but I can only assume that it was sent by Sandro Caparrini, who has a skill for finding rare stories like this. Thank you, Sandro!

[April 2007]: A blog entry was recently posted that comments on my review above of Cantor's War. (I noticed it because this entry was suddenly getting a lot of hits!) You can see what it has to add here.

[May 2008]: Another blog entry was posted that similarly hates this story...but this one is different because the author apparently has an anti-math bias himself and was hoping that the story would successfully make Phipps into a hateful fool. However, recognizing that the story screws up the math ruined it even for him. See Jimbo Jones' comments.