Satan, Cantor, and Infinity Raymond M. Smullyan The College Mathematics Journal, Vol. 16, No. 2. (Mar., 1985), pp. 118-121. Stable URL: http://links.jstor.org/sici?sici=0746-8342%28198503%2916%3A2%3C118%3ASCAI%3E2.0.CO%3B2-U The College Mathematics Journal is currently published by Mathematical Association of America.
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Satan, Cantor, and Infinity Raymond M. Smullyan Raymond Smullyan received his B.S. from the University of Chicago in 1955 and his Ph.D. from Princeton in 1959. His special areas of interest are logic and recursion theory, about which he has numerous publications. A small sample of his authored books include: Theory of Formal Systems (1961), First Order Logic (1968), The Tao is Silent (1978), What is the Name of This Book? (1978), The Chess Mysteries of Sherlock Holmes (1979), This Book Needs No Title (1980), The Chess Mysteries of the Arabian Nights (1981), The Lady or the Tiger (1982), Alice in Puzzleland (1982), 500 B.C. and Other Philosophical Phantasies (1983). Professor Smullyan has been a professional pianist. He is an amateur astronomer with expertise in making telescopes, and he worked his way through college as a professsional magician.
"I have been having a lot of fun with some of our victims," said Satan to Beelzebub, as he rubbed his hands in glee. "In each case I tell the victim that I am thinking of one and only one object out of an infinite set of objects. Each day the victim is allowed one and only one guess as to what the object could be. If and when he guesses it, he goes free. This is the general format of these tests. In some cases the victim has been clever enough tc devise a strategy to win his freedom, in other cases, not. Well, tomorrow I am expecting a new victim and I will arrange matters so he can never go free!" "What will you do?" asked Beelzebub. "I have written down the name of a set of positive whole numbers," said Satan. "Each day he will be allowed to name one and only one set and if he ever names my set, he can go free. But he will never go free!" said Satan shrieking with delight. "Why?" asked Beelzebub.
"Well just look at what I have written!" said Satan.
THE SET O F ALL NUMBERS n SUCH THAT n DOES NOT BELONG
I
TO THE SET NAMED ON THE nth DAY.
I
"I don't understand!" said Beelzebub. "I thought you wouldn't, blockhead! He can't possibly name my set on any day, because for each positive integer n, the set he names on the nth day is different from my set, since one of these two sets contains the number n and the other doesn't! It's as simple as that!" "This sounds like fun!" said Beelzebub.
Well, it so happened that the next victim was a prize student of Georg Cantor! He not only knew his mathematics of infinity perfectly, but was also an expert in semantics and law. In fact he had originally planned to go into law, until he fell under Cantor's magnetic sway and decided instead to go into logic and mathematics. "Before I sign any contract," said the student to Satan, "I want to be sure that I'm absolutely clear as to the terms." "I've already told you," said Satan, "that I have written down the name of some set of positive integers and it is right here in this envelope with my Royal Seal. Each day you are allowed to name one and only one set. If and when you name the set written here, you go free. It's as simple as that!" "I already understood that," replied the student, "but there are several questions that need to be answered. First, suppose that on a given day I name the same set that you have written, but my description of the set is different from yours. Any set can be described in many different ways. For example, suppose you have written: "The set whose only member is the number 2," but on some day I say: "The set of all even prime numbers." Now, the two sets are really the same, since 2 is the onll, even prime number. Yet the descriptions are quite different. What happens then?" "Oh, in that case you win," replied Satan. "I do not demand that our descriptions be the same, but only that they describe the same set." "But that raises a serious problem!" said the student. "It is not always a simple matter to determine whether two descriptions name the same set. Suppose on a certain day I name a set and you reply: "No, that's not the set I have in mind," but I have reason to believe it really is the same set, only you have described it differently. What happens then?" "In that case," said Satan, "you are allowed to challenge me. Now, a challenge is a very serious matter and you should think twice before making it. It might win you instant freedom, or it might doom you here forever!" "Just what do you mean by a challenge?'asked the student. "You challenge me to open the envelope and show you what I have written. If you can prove that the two descriptions-yours and mine-are really of the same set, you win the challenge and go free. But if I can prove that the descriptions name different sets, then you have lost the challenge and your right to name any more sets in the future is cancelled. There is then no way you can ever escape. Remember well that after a challenge, you are not allowed to name any more sets." "That's clear enough," said the student, "but now comes a second point. How do I know that you have really written the name of a set in this envelope?" "You doubt my word?'asked Satan. "Oh, not at all; I don't doubt for a moment that you have written something in this envelope which you believe to be a genuine description of a set, but it has happened in the history of mathematics that what at first sight appeared to be a genuine description has turned out not to describe any set at all-in other words, that there really is no set answering such a description. Such "descriptions" are what logicians call pseudo-descriptions. They appear to describe a set, but really don't. Now suppose that at a certain stage, I have reason to suspect that what you have written in the envelope is not a genuine description, but only a pseudodescription. What happens then?" "If on any day you suspect that I have written only a pseudo-description," replied Satan, "then again you may challenge me. I open the envelope and show you what I have written. If you can prove that it is only a pseudo-description, you win the challenge and go free. But if I can prove that it really is a genuine
description, then you lose the challenge and again your right to name any more sets in the future is cancelled. I must earnestly remind you that after a challenge, you may not name any more sets." "That point is now clear," said the student. "One last thing: Are you willing to have it written in the contract that if at any time you violate any of the conditions, then I go free?" "Yes," replied Satan, "provided that you are willing to have it written that if at any time you violate any of the conditions, then you stay here forever." "Agreed!" said the student. The contract was then drawn up by Beelzebub and duly executed by both parties. "Good!" said Satan. "When would you like to begin?" "Today's as good a day as any," said the student. Let this be the first day of the test. "Very well then, name a set of positive integers!" "The set of all n such that n does not belong to the set I name on the nth day," said the student. "And now I challenge you to open the envelope." "Good grief!" cried Satan, "I never thought of that!" "Open the envelope!" demanded the student. Satan had to open the envelope, and of course he had written the same thing. "So I go free!" said the student. "Not so fast, young man!" said Satan. "You have not really named a set; you have done just what you accused me of possibly doing; you have given only a pseudo-description, not a genuine description!" "Why?" asked the student. "Because the assumption that you have named a set leads to a logical contradiction: Suppose you have named a set. Then this set is the set you have named on the first day. Now, the number 1 belongs to this set if and only if it doesn't belong to the set named on the first day, but since this set is the set named on the first day, then 1 belongs to this set if and only if it doesn't. This is a clear contradiction and the only way out of the contradiction is that you have not really named a set." "I'm glad you realize this," said the student, "because by the same token, you have failed to name a set." "Now, just a minute!" said Satan. "The genuineness of my description is predicated on the assumption that you name one and only one set each day, as you're supposed to. So far, you have not yet named a set today, so I now command you to name a set." "Oh, I have no intention of naming any sets today." "What?'cried Satan, unable to believe his ears. "It doesn't say anywhere in the contract that I must name a set on each day; it says that on each day I am allowed to name a set. Well, it so happens that today I don't choose to name any set." "Oh, really!" shrieked Satan; "you refuse to name a set today, eh? Well I'll force you to name a set today, and tomorrow I'll again force you to name a set, and the day after and the day after, and so on throughout all eternity. You have no idea how terrible my methods are!" "Oh, you can't do that," said the student, "I've already challenged you and it says quite explicitly in the contract that after a challenge, I'm not allowed to name any more sets." Epilogue-Of course. Satan had to set the student free. The student immediately ascended to paradise and embraced his beloved master Georg Cantor. The two had a delightful chuckle over the entire affair. 120
"You realize," said Cantor, "that you didn't have to be this elaborate; you didn't have to start the procedure by giving a pseudo-description. You could have started by simply saying: "I challenge you!" After the challenge, you wouldn't be allowed to name any sets, which would automatically make Satan's 'description' a mere pseudo-description." "Oh, I realized that," said the student, "I just thought I'd have a little fun with him." Discussion and Moral-Satan used Cantor's famous diagonal device to prove that the power set of a set N has higher cardinality than N (see [I], pp. 14, 15 for a proof of Cantor's theorem, or [2], pp. 220-224, for a popular, though equally rigorous version). The student, of course, rightfully guessed that Satan would try and pull such a Cantorian trick. Several people have asked me whether the expression: "The set of all n such that n does not belong to the set named on day n" is a genuine description or not. The answer is that it is a genuine description if and only if on each day there is one and only one set named on that day. If the student would fail to name a set on so much as one day, that would be enough to nullify the meaningfulness of Satan's description. Or if the student would name more than one set on a given day, that would also invalidate Satan's description. But if the student names one and only one set on each day, then Satan's description is perfectly well defined. [A curious thing, though, about this description is that after no finite number of days, can it be known that Satan wrote a genuine description, unless it could somehow be known that the student would name one and only one set each day.] Satan really made a very poor contract! If he had required (instead of just allowed) the student to name one and only one set each day and if he had just deleted this business about the student not being allowed to name any more sets after a challenge, he would obviously have won. Had he done that, then it would indeed have been logically impossible for the student ever to name Satan's set. But as the contract stands, a mere challenge on the part of the student disallows him to name any more sets, which in turn nullifies the genuiness of Satan's "description." The moral of the story is that even fallen angels might benefit from a good course in mathematical logic.
"I'm beginning to understand eternity, but infinity is still beyond me." REFERENCES 1. Stephen C. Kleene, Introduction to Metamathematics, D. Van Nostrand Company, Inc., Princeton, NJ, 1952. 2. Raymond M. Smullyan, What Is The Name Of This Book?, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1978.
