Chapter 10 EXTRA: Dr. Linda teaches Paul the Russell Paradox
...... Paul went to Dr. Linda's office, knocked,
and was admitted. Dr. Linda, chalk in hand, stood in front of a blackboard filled
with symbols--like a written incantation of magic. Paul was envious; he wished
he understood that magic.
...... Dr. Linda had Paul sit and then, still holding her stick of chalk, she sat facing him. "So, Ralph gave you the hangman's paradox. Yes?"
...... Paul nodded. "I'm not too worried about it, though. It seems to have more to do with language than math. Maybe it's something to do with the meaning of surprise."
...... "It does seem to have something to do with language," said Dr. Linda. "But don't underestimate it." She took a file card from her desk and wrote something on each side of it, and then handed it to Paul. On one side of the card was written,
...... ...... ...... What is written on the other side of the card is false.
...... Paul turned the card over and found that
the same thing was written on the other side.
...... "This is a shorter paradox," said Dr. Linda. "Also related to language. If you think about it, you'll see that what's written on the card is neither true or false."
...... Paul started to think about it.
...... "But think about it later." Dr. Linda took the file card out of Paul's hand and slipped it into his shirt pocket. "There is a paradox though, an important one, that isn't about language--not the usual way we think about language, anyway." She leaned back in her chair. "All right then, the Russell Paradox." She gazed at him for a few seconds and then said, "Paul, do you know the term 'set' meaning a collection of things?"
...... "And something in the set is called a member of the set?"
...... Paul nodded and opened his notebook.
...... "You won't need to write anything down," said Dr. Linda. "For me, anyway, writing things down interferes with my thinking about them. And I just need you to think."
...... Paul closed the notebook.
...... "Good." Dr. Linda stood and went to the board. She didn't write anything on it, but just seemed more comfortable standing in front of it with a piece of chalk in her hand. "Okay then. Imagine the set of all...of all cats."
...... Paul visualized a football stadium filled with squirming, meowing cats.
...... "Is the set of all cats a cat?" said Dr. Linda.
...... "What? You mean are all the cats in the universe a single cat?" Paul wrinkled his nose. "No."
...... "Good," said Dr. Linda. "The set of all cats is not a cat. Now how about the set of all concepts? Is that a concept?"
...... "The set of all concepts? You mean, is the set of all concepts itself a concept?"
...... Dr. Linda nodded. "Yes."
...... "Well, I guess it is."
...... "It is," said Dr. Linda. "The set of all concepts is in itself a concept." She looked quizzically at him. "Are you comfortable with that idea?"
...... He nodded. "Yeah, okay."
...... She took a step toward him. "Now this is important: since the set of all concepts is itself a concept, the set of all concepts includes itself." She gazed inquisitively at Paul. "Do you understand that? One of the members of the set of all concepts is--the set of all concepts."
...... Paul bit his lip.
...... "Take your time. Think about it." She chuckled. "This is one of those ideas that sound complicated when you first hear it. But when you do think about it for awhile, it becomes trivial."
...... "Yeah." Paul played with the notion. "Yeah. The set of all concepts is, like, a concept, so the set of all concepts has to be in the set of all concepts."
...... "Good," said Dr. Linda. "We call sets that include themselves, self-inclusive sets. Okay?"
...... Paul smiled. "Okay."
...... "The sets that don't include themselves," Dr. Linda went on, "are called non-self-inclusive sets. Like the set of all cats. That's a non-self-inclusive set."
...... Non-self-inclusive set. Paul really enjoyed playing with these complex-sounding terms--and understanding them.
...... "So, here's the big one," said Dr.Linda. "Imagine the set of all non-self-inclusive sets."
...... "Oh, boy!" Paul wrinkled his nose. "This is hard."
...... "It's the only hard part." Dr. Linda returned to her desk and sat. "Just think about it for a minute--until you get it. Until it feels obvious."
...... Paul closed his eyes for a few seconds. "Yeah, okay. The set of all non-self-inclusive sets. Got it. The set where the members are all cats, all candy bars, all math books, all pickles, and like that."
...... "Exactly," said Dr. Linda. "Now, this is the question. Is that set self-inclusive?"
...... After a moment, Paul said, "It can't be,"--He felt proud of himself for following an argument with big words--"since it has only non-self-inclusive sets inside of it."
...... "Smart boy!" Dr. Linda smiled warmly at him. "Good thinking. But, and this is a very big but, if the set of all non-self-inclusive sets is non-self-inclusive, then, by definition, it has to be in the set of non-self-inclusive sets. Yes?"
...... Paul wrinkled his nose again and thought about it. Then, into the silence, he gave a soft "Wow!"
...... "Neat, isn't it?" Dr. Linda slapped down the chalk on the desk. "All right. We've proved that the set of non-self-inclusive sets can't be self-inclusive and can't be non-self-inclusive. That is a heavy-duty paradox."
...... "What's the answer to it?"
...... "Answer?" Dr. Linda spread her hands. "I don't know. There might not be one."
...... "But... But there's got to be."
...... "Because...because mathematics is perfect. There shouldn't be any paradoxes in mathematical logic."
...... "I used to think that way," said Dr. Linda in a distant voice. "Mathematics was my religion, too. But then I found out about..."
...... "The Russell Paradox?" Paul supplied after a couple of seconds.
...... "No," she said. "Something else." She paused. "I don't think you're ready for it yet."
...... Paul felt a surge of resentment. He felt he was being treated like a kid. "When will I be ready for it?"
...... She stared clinically at him for a moment, then said, "this evening. Let's have another meeting, a short one, after dinner. Okay?"
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