"Cryptonomicon" - читать интересную книгу автора (Neal Stephenson - Cryptonomicon (The Whole Book))

"I have used the Leibniz series in order to placate our friend. See, Lawrence? It is a string of symbols."

"Okay. I see the string of symbols," Lawrence said reluctantly.

"Can we move on? GЎdel said, just a few years ago, 'Say! If you buy into this business about mathematics being just strings of symbols, guess what?' And he pointed out that any string of symbols--such as this very formula, here--can be translated into integers."

"How?"

"Nothing fancy, Lawrence--it's just simple encryption. Arbitrary. The number '538' might be written down instead of this great ugly [sigma], and so on.

"Seems pretty close to wanking, now."

"No, no. Because then GЎdel sprang the trap! Formulas can act on numbers, right?"

"Sure. Like 2x."

"Yes. You can substitute any number for x and the formula 2x will double it. But if another mathematical formula, such as this one right here, for calculating pi, can be encoded as a number, then you can have another formula act on it. Formulas acting on formulas!"

"Is that all?"

"No. Then he showed, really through a very simple argument, that if formulas really can refer to themselves, it's possible to write one down saying 'this statement cannot be proved.' Which was tremendously startling to Hilbert and everyone else, who expected the opposite result."

"Have you mentioned this Hilbert guy before?"

"No, he is new to this discussion, Lawrence."

"Who is he?"

"A man who asks difficult questions. He asked a whole list of them once. GЎdel answered one of them."

"And T№ring answered another," Rudy said.

"Who's that?"

"It's me," Alan said. "But Rudy's joking. 'Turing' doesn't really have an umlaut in it."

"He's going to have an umlaut in him later tonight," Rudy said, looking at Alan in a way that, in retrospect, years later, Lawrence would understand to have been smoldering.

"Well, don't keep me in suspense. Which one of his questions did you answer?"

"The Entscheidungsproblem," Rudy said.

"Meaning?"

Alan explained, "Hilbert wanted to know whether any given statement could, in principle, be found true or false."

"But after GЎdel got finished, it changed," Rudy pointed out. "That's true--after GЎdel it became 'Can we determine whether any given statement is provable or non-provable?' In other words, is there some sort of mechanical process we could use to separate the provable statements from the nonprovable ones?"