Satan, Cantor, and Infinity and Other Mind-Bogglin by Raymond M. Smullyan

Author:Raymond M. Smullyan
Language: eng
Format: mobi
Tags: Science, Philosophy
ISBN: 9780307819826
Publisher: Knopf
Published: 1992-01-01T00:00:00+00:00

II. CAN YOU FIND THE KEY?

“If you like curiosities and complex problems,” said Quincy, “I must tell you of a curious system devised by Professor Cudworth, who is another roboteer on this island.

“He was curious to know what would happen with my system if the normalization rule—Rule N, which is really at the heart of my system—was replaced by the repetition rule—Rule R of Roberts. He tried this and got nowhere. Yes, he could solve a few trivial problems, but they were useless for constructing programs. Indeed, he couldn’t even get a program for a self-duplicating robot—that is, he couldn’t even get an x such that x creates x. Then, a friend of his, Inspector Craig of Scotland Yard, who takes a great interest in combinatorial puzzles suggested that Cudworth add the reversal rule, and when he did so, the system worked fine! It could do everything that could be done in my system, and also everything that could be done in Roberts’s system, and more!”

“What is the reversal rule?” asked Alexander.

“Oh, by the reverse of an expression is meant the expression written backwards. For example, the reverse of ABCD is DCBA. Well, Cudworth took the letter V and added the rule that if x names y, then Vx names the reverse of y. Then, as I said, the system worked perfectly.”

Quincy then reviewed the rules of Cudworth’s system for his three guests:

Rule Q1Q2. Q1xQ2 names x.

Rule R. If x names y, then Rx names yy.

Rule M. If x names y, then Mx names Q1yQ2.

Rule K. If x names y, then Kx names y#.

Rule V. If x names y, then Vx names the reverse of y.

“The creation, destruction, friendship and enmity rules are the same as in my system,” continued Quincy. “Now, the first problem is to find an x that names itself. This is not simple! The shortest one I know has eighteen letters. There is also an x that names its own norm. The shortest one that I know has thirty-four letters.

“There is also an x that names its own reverse, one that creates its own reverse, one that destroys the reverse of its repeat, one that is a friend of an enemy of the reverse of the norm of its own quotation, and I could go on endlessly mentioning other combinations. The system also obeys the fixed-point principle, the double fixed-point principle, and so forth.

“You might have fun working these problems out at your leisure. I believe that once you solve the first problem, you will have the key to all the others.”

• 19 •

Professor Quincy was right in saying that the essential idea behind the construction of an x that names itself is really the key to solving all the others. What is the key?

When our three friends left Professor Quincy, they felt that it was too late to set sail for home that evening, so they decided to stay overnight at an inn.

“Tomorrow,” said the Sorcerer, “before we leave this island, I would like us

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