'It's about the Daedalus?'
'It's about why my brother was killed.'
'And you think an alumni was involved?' Cho asked.
'It's a long shot.' Gideon shook his head. 'I really don't know what I'm doing here, but I can't stop looking for a reason. Something . . .'
'I understand.'
Gideon shook his head and started to stand. 'No, you don't have to talk to me. I'm not a cop right now, and I shouldn't even be here—'
'No' Cho said. He wheeled up and placed a hand on Gideon's cast, easing him back into the chair. 'It's all right.'
When Gideon was seated, Cho rolled back and said, 'I know what it's like to go through something like that, to hunt for a meaning—but I should warn you, no one outside yourself can tell you why it happened. The most they can do is tell you how.'
Gideon nodded. 'This man, I believe he might have hired the driver for the Daedalus.'
'Part of the terrorists the Secret Service was supposed to capture?'
Gideon nodded, though he wondered—if the ambush wasn't really Secret Service, what did that mean about the people they had meant to ambush.
'So who is he?' Cho asked.
'About six feet, two hundred pounds. Mid-twenties, white, blonde, named Mike.'
Cho waited for a long time before he said, 'Is that it?'
Gideon nodded.
'Well good luck. You know how many people go through MIT, and you don't even have a specific department to look in.' Cho exhaled, 'If you had a graduation date, or a year he was here, or even a less common name.'
'I know,' Gideon said. 'I think I must have been half nuts coming all the way from D.C. This Mike had on an MIT sweatshirt—'
'That's it?'
'All I had to go on, yes.' Gideon nodded. 'But that's only half the reason I'm here. This, you might actually be able to help me with.'
' We'll see. . .'
Gideon handed over his paper. 'I've been trying to find out what this means. A friend of mine said it looked like a mathematical symbol.'
Cho looked at the page for a few moments, as if deciphering Gideon's handwriting. Eventually, he nodded. 'Your friend was right. What you have here is aleph-null.'
Up to now, Gideon had been convincing himself that this whole trip was a waste. Despite that, when Cho identified the name of the symbol, Gideon felt a thrill. He had found something.
It meant something . . .
'What is it?' Gideon asked, his voice was rushed and breathless.
'It's the lowest class of infinity.'
Gideon sat back, frowning. That made no sense to him.
'You've never heard of it, have you?' Cho asked.
Gideon shook his head, feeling as if the answer was just as cryptic as the unexplained symbol. It sounded like some New Age bullshit to him. 'I thought infinity was infinity.'
Cho handed back the paper and said. 'That's a logical way to look at it. It seems so intuitively obvious, that the man who discovered—or invented, depending on how you look at it—the transfinite numbers was probably the least appreciated mathematical genius of the past five hundred years.'
It was still all Zen to Gideon. 'What does the lowest class of infinity mean?'
Cho leaned back and steepled his fingers a moment. 'To explain that I'll have to give you some very rudimentary set theory—'
Gideon didn't like where this was going. 'The last mathematics I had was trig in high school.'
'Don't worry, this is fairly simple.' He reached into the piles of papers on his desk and started dropping things in Gideon's lap—paper clips, rubber bands, pencils . . .
'Hey.'
'There,' Cho said, having placed three piles in Gideon's lap.
'What are you doing?'
'Some people need help visualizing.' He smiled. 'Now we have here three sets. For the sake of example, these piles are infinite.'
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.'
