Gideon looked down and said, slowly, "Okay . . ."
Cho picked up a paper clip and Gideon's leg twitched involuntarily. "We'll call the paper clips Set 'Z,' and the rubber bands set 'P,' and pencils set 'N' . . ."
"Whatever you say."
"How do we know we have the same number of pencils, rubber bands, and paper clips." Or, in set theory, how do we prove that set N is equivalent to sets P and Z, and vice versa?"
"We just count how many—"
Cho shook his head. "These are infinite piles, counting is not allowed." He placed the paper clip back on Gideon's leg. "There's a simple solution. Ask yourself how you would make equal piles of finite objects without 'counting' them."
Gideon looked down at the piles on his legs. How the hell could you measure the piles without counting them? Gideon reached down with his good hand and arranged the groups in rows, trying to think of what Cho was getting at.
It seemed stupid, he didn't even have equal piles. There was an extra rubber band that stuck out once he aligned the groups . . .
"Wait a minute . . ." Gideon looked down at his legs, the beginning of a realization coming to him.
"Yes?" Cho asked.
"When you align them," Gideon picked up the extra rubber band. "You can see when they aren't equal."
Cho nodded. On Gideon's leg, the groups sat in neat rows, each pencil next to a rubber band and a paper clip. "What you just did was put everything into a one-to-one relationship with each other. Finite sets are equivalent when each member in one set can be paired with exactly one element in the other set, with no leftovers." He took the extra rubber band. "Obviously, our finite set of rubber bands wasn't equivalent to the others."
"But we were talking about infinite sets."
Cho nodded. "Now, though, you have a way to measure the equivalency of our infinite piles of paper clips, rubber bands, and pencils."
"Okay, I can see that." Gideon could imagine mountainous piles of rubber bands, pencils, and paper clips, and going to each one, and taking one item per pile and grouping them together. He could do it forever, and no pile would come up short. "But I don't see how that keeps infinity from being infinity."
"Bear with me. We have our infinite sets. Now let's take our pencils, set 'N,'. We can give each pencil in our pile, one at a time, a number. One, two three . . ."
"Okay, you can go on forever."
Cho picked up the pencils and said, "Having done that mental exercise, we have a pile of the natural numbers."
"So an infinite pile of pencils is equivalent to the set of natural numbers."
Cho nodded. "Considering the pile as a set of infinite individual pencils. Now set P—we'll number each rubber band with a successively higher prime number." He picked up each rubber band as he spoke. "Two, Three, Five, seven—"
"That goes on forever, too?"
Cho nodded and picked up the paper clips. "Set Z. Zero, One, Minus One, Two, Minus Two."
"I have the picture I think." Gideon said. "The set of natural numbers is equivalent to the set of prime numbers—"
"—is equivalent to the set of integers."
"If you say so."
"That equivalence was a problem that George Cantor set out to solve. I've just explained the basics of the proof. The set of natural numbers 'N,' is equivalent to the set of primes 'P,' the set of integers 'Z,' and even to the set of all fractions. Aleph-null is the number of elements in those sets."
"Why the symbol, then? If all infinite sets are equivalent to each other—"
"They aren't," Cho said.
"How? I mean if you have the numbers from one to whatever, you can pair them up with a series of anything."
"Not quite."
Cho dropped his piles of objects on the desk and grabbed a pad of graph paper. "This is harder to describe without using actual numbers." He took one of the infinity pencils and scribbled for a moment. After a few moments he handed the pad to Gideon.
Gideon stared at the page.
Gideon looked at it for a long time. He could see the first three sets as what Cho had described to him, the arrows emphasized the one-to-one relationship Cho had talked about.
The fourth one was different. The others trailed off to the right, Gideon could figure out that's what the dots meant. But there were dots between the numbers in set R. "You're saying that there's an infinity between each of these numbers, right?"
"That's a good way to put it. That's the set of reals. Once you add irrational numbers to the number system, there's no systematic way to pair every element with the set of natural numbers. Any method you try will have elements—an infinity of elements—slipping through the cracks."
"So aleph-null is the number of elements just in the first three sets?"
Cho nodded. "It's the size of the set of natural numbers, the set of primes, the set of rational numbers . . . The set of real numbers is much larger. It's aleph-null raised to the power of aleph-null."
"I'll take your word about that. I think I follow you enough to answer my question." Answered and not answered . . . Gideon still wasn't sure what it meant that these people were using an esoteric symbol for infinity.
"Why are you interested in this, by the way? It seems to be a bit far afield for police work, even if you are working on your own."
"I needed to find out what it meant, to discover why a group of people might be using it."
"Using it? How?"
"As sort of a logo."
"Logo—"
"It was spray-painted on a wall near the Daedalus."
"Now that is a strange piece of graffiti."
"It was also on a business card. I thought that discovering its meaning would give me a clue to what the group was, and who's a part of it."
Cho leaned back, "A logo, hmmm . . ."
Gideon nodded. "This tells me something, though. The group that uses this as a symbol probably wants the Daedalus for a different reason than one that'd use a religious or occult symbol . . ." For some reason, the image of the tarot card, the Fool, flashed through Gideon's mind. The symbol of infinity had its own connotations that could range from the scientific to the theological.
As if echoing Gideon's thought, Cho said, "Some might call mathematics a religion . . ."
He rolled around and rummaged in a filing cabinet. After a bit, he pulled out a folded stack of papers, stapled together. He shook them so the individual pages separated. "Using aleph-null as a logo, hmmm."
He handed over the papers. The first thing that caught
Gideon's eye was the familiar " N' taking up a large section of the top left corner of the first page. The paper was titled, "Aleph-Nulclass="underline" The Newsletter of the Evolutionary Theorems Research Lab."
Gideon looked up and asked, "What's this?"
"The one group I know of that used aleph-null, like you said, as a logo. The ET lab isn't around anymore, but it carried on research for about four years or so."
"What was it?"
Cho shook his head. "I think you should talk to some Comp-Sci people about that. I know they were supposedly doing original work in number theory, and there was some controversy about some of their proofs before Dr. Zimmerman left."
"Dr. Zimmerman?" Gideon borrowed Cho's pad and pencil and wrote down the name under the collection of sets that Cho had written down.
"Yes, Julia Zimmerman, she headed the ET lab—invented it."
Gideon tore off the sheet of graph paper and pocketed it. "What do you mean she left?"
"After the ET lab was shut down, she resigned. I don't know where she went to from here. But I gather she had a rather confrontational reputation, which wouldn't have helped her find another post."
"What prompted the reputation?"
Cho shook his head. "I don't know much about it. It was all academic politics and it happened while I was still a TA. Go check with some Comp-Sci people. Folks with tenure."
"Uh-huh." Gideon folded the newsletter and asked, "Can I keep this?"
Cho nodded. "It's yours." He took Gideon's hand. "I hope you find what you need.