*and*curmudgeonly hatred of math dorkery. If this sounds like a bad time, skip to the next post on your friends page now. Kthxsorry.

Still here? OK. But don't say I didn't warn you.

*Cantor's War*is an SF story written by Christopher Anvil in 1974. It recently got reissued in a Baen anthology, and I read it and remembered it from my much younger days. The plot, basically, is as follows:

1. Aliens are attacking from "tau space", in which any number of ships are infinitely multiplied (but divided back into their original number before they can leave "tau space".)

2. This one dude who has a doctorate in math is really fucking irritating and condescending.

3. Cantor's Theorem is dead wrong, ha ha, let's laugh at the irritating math PhD shithead.

Now, in my younger days, I

*loved*this story. I want to love it still, because honestly... yeah, I kinda do know several of that irritating math dude, and yeah, I still get extremely annoyed with him. But the problem is, Anvil desperately failed to understand the most basic principles of set theory when he laid out the story.

Mathematically, here is his premise, slightly expanded:

1. human ships can outfight alien ships on a 1:1 basis

2. aliens can put more ships into "Tau space" than humans can

3. "Tau space" can best be visualized as infinitely kaleidoscopic, such that any unit which enters "tau space" will be reflected in an infinite number of "sectors" throughout; a single grain of sand will be mirrored once in each "sector", whereas five grains of sand will

*each*be mirrored once in each sector

4. even though "Tau space" multiplies any number of ships infinitely, the aliens have a "denser infinity" than the humans do since they put more ships in, so they win

This gets worked in the plot by the irritating math PhD character pointing out that the series 2,4,6,8... and the series 1,2,3,4... are each infinite, and infinity == infinity, therefore the humans can overcome their "density deficit" in "tau space" by simply moving ships from the end of the series toward the beginning until they attain parity. This appears to work at first, but then the aliens start doing the same thing, and they quickly overwhelm the humans (who only entered a single ship into "tau space").

Now, the thing is, honestly I can understand and even support some math geek hate. (Sorry mathies. I love some of you. But ... yeah.) And the irony is, Anvil

*could have pulled it off*with the way he laid his story out so far. One way Anvil could have done so was to have his main character, a mary-sue-ishly competent military man who coolly cuts the mathy scapegoat down to size, explain that the problem is that it took

*a finite amount of time*to move each ship replica to a different sector of "tau-space". This would have meant that since the aliens had more ships in each sector of "tau-space" to begin with, they could concentrate force faster than the humans in any given area, thus resulting in inevitable defeat. This would have the virtue of being perfectly correct. It would also serve well as an example of a likely way that an irritatingly self-absorbed and condescending mathy type might miss a real-world problem in his initial - and technically correct - statement of a principle.

BUT, unfortunately, that's not what Anvil did. He instead had his mary-sue make a smug analogy in which the humans had an infinite number of ships, only half of which were piloted, and the aliens have an infinite number of ships, all of which were piloted. He then said that irritating mathy guy only compared the

*left*end of his series 2,4,6,8... and 1,2,3,4... and that if one shifted pilots on the human side to pilot all the ships, one must necessarily leave more ships unpiloted at the other end, so you couldn't possibly have as many piloted ships as the aliens. QED, bang, shit didn't work, you're stupid, mathy guy. Unfortunately, all this REALLY shows is that Anvil doesn't understand the difference between cardinality (the number of items in a set) and ordinality (the way you

*label*individual items

*in*a set).

The

*cardinality*of the two sets 1,2,3,4... and 2,4,6,8... is

*equal*. There are

*precisely as many items*in each of the above sets - in fact, one intuitively more-accessible-to-the-layman way to look at it is that they are precisely the

*same*set of items, merely with

*different labels on them*. How can we demonstrate this? Well, let's decide to arbitrarily re-label the individual items of the set 2,4,6,8... by tacking a "/2", or "divided by 2", onto the end of each. Now the set 2,4,6,8... reads 2/2,4/2,6/2,8/2...

*which is equivalent to*1,2,3,4... if you parse the "label". So, clearly,

*the cardinality of the two sets*1,2,3,4... and 2,4,6,8... is equal, since you can make the sets themselves equivalent just by changing the label - the

*ordinality*- of either set.

So, yeah. I desperately would like to love that story. I have far more sympathy with the military officer than with the mathy. But damn it, the author just got it WRONG and that ruins it. Worse, he could SO EASILY HAVE MADE THE DAMN STORY WORK without getting the math wrong. Aside from the already-mentioned finite-transit-time problem, I can easily envision an alternate scenario in which the snooty math guy just tells the colonel, "my dear fellow, you must merely change the ordinality of your set of ships and all will be well" without coming up with any valid way of doing so. Or if you prefer a more humorous approach, the mathy character could be had to just scrawl "blue: x10 scale" on the monitor in Magic Marker and say it's all fixed, and get punched out for it. But, no. Instead, tortured, triumphant, and WRONG analogy.

So in the end, Anvil made the type that I identify with - the engineer or applied scientist - look ignorant by his own botched attempt to make fun of the pure scientist counterpart. And that's what REALLY pissed me off about the story.

As a post script, it turns out I'm not the only one who ranted about this story, though I do at first glance appear to be the only complainer who actually

*wanted*the plot (sneering mathy gets his comeuppance) to be done well. Huh.

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