'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-Null: 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.
1.13 Mon. Mar 9
M ORRIS Kendal leaned back in his office chair watching a video that C-Span had sent him. He watched a half-dozen men brought before a federal judge and arraigned for the hijacking of a Daedalus supercomputer. The tape dated from about forty-eight hours after the Secret Service fiasco. Up until then, the men had been held without bond or access to a lawyer.
There were rumors that Congress might offer some of them—the ones not directly involved in the shooting of a state trooper—immunity for their testimony in their closed-session investigation of the Secret Service screw-up. Kendal hoped the rumors were unfounded.
Even if they were, Gideon was right. Something wasn't kosher here.
The tape didn't show anything that wasn't public record. The various news agencies, from CNN to Hard Copy had already done the expose of these men. They were a group of 'security consultants.' Despite the title, they had little in common with Kendal. These men were all Nicaraguan and Salvadoran ex-military. Applied to them, 'security consultant' was a euphemism for hired thug.
Anyone familiar with the death squads of the eighties would have recognized the name of Colonel Luiz Ramon before this incident; now everyone who followed the news would know that since then he'd been implicated in things ranging from drug smuggling to assassination.
'Who hired you boys?' Kendal asked the screen.
Kendal had sources, and knew slightly more than the news agencies did. He had the Colonel's statement to the Justice Department—which he had received in a manila envelope passed to him in the Capitol Rotunda by a very nervous secretary who had owed him a favor. It wasn't very enlightening. The Colonel wasn't saying anything.
It also seemed as if the main defense would be the violation of Luiz Ramon's rights during his capture and the two weeks after. The Justice statement had been given, pointedly, after the period of black captivity before the whole episode blew up in the news. If the Colonel had said anything before that point, Justice didn't have a record of it.
Justice didn't even have a record of where the Colonel had been held before Gideon was shot.
Justice did have a record of a Costa Rican bank that received a wire transfer from a District bank shortly before the hijacking. Two million dollars, paid for with a cashier's check.
For that much, Colonel Ramon should've run the operation a little less as if he was still goose-stepping in some third-world banana republic. He wondered if the Colonel still did the odd job for the CIA, like he had when he
