The Library Paradox (17 page)

His tone subtly implied that there was no possible comparison in importance between philosophy and mere ‘matrix theories’. I asked myself why Professor Taylor had insisted on my being introduced to this person, and concluded that he must have known Professor Ralston. I wondered briefly how I could lead the subject of conversation in that direction before the young man decided that I was not worth his notice.

‘And here is Dr Burali-Forti,’ Professor Taylor continued, drawing forward a rather tall and strong gentleman of a rustic, Mediterranean appearance, who had just approached our group. ‘He is visiting our department from Italy, and Russell has come down here to work with him.’

Professor Hudson and a few other mathematicians had by now collected about us, birds of a feather sticking together.

‘Have you written up anything on your paradox, Burali?’ asked one of them.

‘No, no,’ replied the person addressed, with a strong Italian accent. ‘I have not understand it properly yet. It is a deep and strange thing. It-a does not make much sense, I do not yet see the real importance of it for mathematics. I must think. We are very much talking with Russell about this. He is trying to find ways of saying the same thing differently.’ He smiled at his younger colleague, who frowned.

‘A paradox?’ I said, associating it instantly with my own paradoxical problem.

‘Are you interested in paradoxes?’ he asked me at once, clearly eager to talk about it.

‘Ah, I have heard of some mathematical paradoxes already,’ I stammered. ‘You know what I mean, surely. Those stories that mathematicians use to torment us ordinary souls, like the one explaining that if the hare starts the race after the tortoise, he will never be able to catch up for some reason …’

He laughed. ‘Zeno’s paradox states that the hare – or the fast runner Achilles – will never catch up because whenever he reaches the place where the tortoise just was, the tortoise has already advanced from that place,’ he said. ‘Mine may not be so easy to explain as that one. But your husband is a mathematician? You are, perhaps, used to some mathematical terminology?’

‘Well, yes, if it is not too difficult,’ I admitted, wondering what I was letting myself in for.

‘Then I shall be very happy to explain it to you!’ he said with alacrity. A discreet, dark-clothed being floated by with a tray, and I found myself holding a fluted glass of champagne. Dr Burali-Forti took one as well, and tapped it gently against mine, producing the pleasing ‘ting!’ that only beautiful crystal can yield.

‘Drink, drink,’ he said happily. ‘All definitions will then seem very simple! Now, do you know what a totally ordered set is?’

‘Yes, I do know that,’ I answered, feeling that I was passing an examination. ‘It is just a set which is ordered by some order. I mean, if you take two different elements
a
and
b
in the set, then either
a
is less than
b
or
b
is less than
a
. That means you can line up all the elements in an ordered line, doesn’t it, just like the ordinary numbers.’

‘Yes, very good, very good,’ he said. ‘Now, such a set is also called
well-ordered
if it has a
smallest element
, and every one of its subsets also has a
smallest element.
That is not so easy to see, is it? The ordinary counting numbers 0, 1, 2, obviously form a well-ordered set.’

‘Yes, I quite see that,’ I concurred.

‘Well, what you now need to know, and this is where it becomes conceptual, is that
every well-ordered set has a unique ordinal number.
If the set is finite, all is well, the ordinal number is just the number of elements of the set. But if the set is infinite? Wait – do not say that the ordinal number is then infinity! In this theory, there are no
infinite numbers.
If a set is finite and contains ten elements, its ordinal number is 10. If the set is empty, its ordinal number is 0. All the counting numbers 0, 1, 2 and so on are thus the ordinal numbers of finite sets. But there are bigger ordinal numbers! For instance, we write
omega
for the ordinal number of the set of all counting numbers. You can see that it is not the same as the ordinal number of the set of points on a line, for instance. God forbid you should say omega equals infinity! All these numbers measure different intensities, different quantities, different
types
of infinity. Every well-ordered set has its ordinal number, and these ordinal numbers themselves form a well-ordered set, with 0, the ordinal of the empty set, as smallest element.’

‘You mean that you classify infinite sets into different
sizes
of infinity,’ I said, catching something of his gist. ‘I do see that one cannot say that there are as many counting numbers as there are points on a line, even though there are an infinite number of both. There seem to be far
fewer
numbers than points.’

‘That is exactly it,’ he concurred, ‘different infinities can be compared. I said that the ordinal numbers form a well-ordered set, so they can be arranged in a row, starting with 0, the smallest one, and going up according to their order. And it is infinite. No matter how big an infinity you have found, there is always a greater one. Do you agree?’

‘Insomuch as such a thing can be conceived,’ I said, trying to visualise an infinitely increasing sequence of increasing infinities of infinity and succeeding only in imagining larger and larger versions of strange letters from foreign alphabets streaming before me.

‘So, the set of these ordinal numbers, measuring the sizes of infinite sets, itself forms a well-ordered set,’ continued the Italian professor with irrepressible aplomb, ‘and thus, like any well-ordered set, it has an ordinal number.’

‘Ye-es.’

‘Then, let us have the paradox!’ he said triumphantly. ‘Start at the left of the row of ordinal numbers, with 0, and take everything along the line to the right as far as you like. Then stop. What you have taken is a subset of the complete, infinite row. This subset is of course also well ordered, just as the full set is.’

‘It would seem so, yes.’

‘But the ordinal number of any subset is greater than any of the ordinal numbers that are in the subset.’

‘Ah, really?’ I said.

‘Yes! for instance, the set {0,1,2} has 3 elements, so its ordinal number is 3, which is greater than any element of the set {0,1,2}.’

‘Oh, I see.’

‘But now, what if the subset you take is the entire set? You start at the left, and move along to the right
all the way.
You include everything! Then you have a subset, which happens to be the full set, and so its ordinal number is greater than any ordinal number which is inside your subset, but your subset
is
the full set of ordinal numbers, so that none can lie outside it.’

Alas, my head was spinning by this time.

‘I wish I could understand it better,’ I said regretfully.

‘You can,’ said the young Bertrand Russell, who had been standing by, listening. ‘Burali – I see what it is I’ve been trying to tell you. This makes it all much simpler! What you’re really saying is that
you can’t have the set of sets, which don’t contain themselves.
Of course! That’s what I’ve been trying to formulate clearly all these last weeks!’

He seemed surprised and amazed at his own remark. Dr Burali-Forti cast him a look of confused admiration.

‘I don’t quite understand what you mean,’ he said honestly.

‘Neither do I!’ I chimed in, probably even much more honestly.

‘But it’s simple,’ he told me. ‘Yet no, you are not a
mathematician. Let me think for a moment.’ He pressed his hand to his forehead for a moment, then looked up, his eyes twinkling.

‘Let me tell it this way, then,’ he said. ‘The story takes place in a country with many libraries. A master cataloguer tells each librarian in his land to make and send him a catalogue listing all the books in his library. But when the master cataloguer receives all the catalogues, he notices that in some cases, the librarian has included the catalogue itself in the list, and in other cases, he has not done so. The master cataloguer now wishes to list the catalogues he has received, and to begin with, he attempts to make a list of
those catalogues which do not list themselves,
by which I mean those which are not included in the list of books they contain. And here is the paradox: should the master list the catalogue he is presently writing amongst those catalogues which do not list themselves, or should he not?’

‘Well, if he does not include his own catalogue in his list, then his catalogue becomes one of those which do not list themselves – oh!’ I said.

‘Exactly,’ he said, lifting his chin. ‘And in that case, he should add his catalogue to the list of those which do not list themselves. But if he does it, then he should not do it, for he cannot include a catalogue, which lists itself, in the list he is presently making. That is really a new way of expressing the paradox!’ He turned to the few people who were standing about us, listening to him, with the air of a successful conjuror.

‘This is a very important discovery,’ said Dr Burali-Forti.

‘It is impressive,’ said Professor Hudson, ‘so easy to explain, and yet so peculiarly contradictory. But now that I see what you are saying, I can tell you something extremely surprising. I myself – a mathematician – did not spot anything extraordinary or important about it until this very moment! But as it happens, I have already heard this very same paradox, albeit in a different version, not such a pleasant one.’

‘Who did you hear it from?’ asked Russell sharply, looking displeased.

‘From someone who is now dead, and who was not even a mathematician,’ said Professor Hudson, with a slightly ironic expression. ‘Your discovery is safe.’

I began to suspect something.

‘Are you referring to Professor Ralston?’ I asked suddenly.

‘Exactly.’

‘Ah,’ said Professor Taylor. ‘You must be thinking of that paradox about Israelites that he was so fond of telling. Are you saying that, logically, what he told is the same thing as this library story?’

‘It’s exactly the same,’ he replied.

‘Let us hear it,’ said Dr Burali-Forti, with scientific interest.

‘Well, I’ll tell it to you,’ said Professor Hudson with some distaste. ‘Please bear in mind that it represents nothing of my own views and that its author has passed away. I am only repeating it because I have just realised that its scientific content is actually much more significant than I would have believed. His idea was to take a, ahem, well, a
specific group of people, ahem. Israelites, for example. Now, take each person in the group, each Israelite as it were, and ask him the following question: does he feel himself as a member of some specific group, or not? Some will respond that yes, they belong to a group. Others will respond that they do not feel they belong to any group. Now, group together all those who reply that no, they do not belong to any group. Pick out any one of them, and ask him if he feels he belongs to that group; that group of non-belongers, as it were. If he says yes, as a non-belonger he is in the group of non-belongers, why, there is your contradiction. He should not be there, since he claims not to belong to any group. But if he insists that he does not belong to any group, then he cannot be in the group of non-belongers, so he must belong to the other group, namely the group of belongers. It is truly the same paradox. Ralston interpreted it as saying that the denial of the original belonging, the denial of the race, in fact, produces a contradiction. That was the aspect of it that he enjoyed; it seemed to say that a person’s identity as an Israelite is not something which can be denied; it is not up to the members to choose. It was his way of saying that assimilation was meaningless. Well, that was the bee in his bonnet.’

‘It’s a little amateurish,’ said Russell. ‘What is really needed is to put the whole reasoning in the context of set theory. I must get to work on this.’

Professor Ralston was a revolting personage, I thought, but did not, of course, speak the words.

At this very moment, dinner was announced. Professor Taylor personally guided me into the dining room, and I found myself seated with him on one side of me and Dr Burali-Forti on the other. Professor Hudson sat across from me, flanked by his family. Emily sat next to Roland Hudson and continued to look pink. I glanced at them once or twice out of the corner of my eye. I thought of Jonathan’s devoted gazes and felt a little sad.

‘So you have heard all about the paradox?’ Professor Taylor asked me as mock turtle soup was served.

‘I believe I have heard more about it than I can possibly digest,’ I replied. ‘I feel I still have a lot to learn,’ I added as significantly as I could, looking at him.

‘Ah, I was forgetting!’ he responded, nodding. ‘I must tell you the most important piece of news. Bernard Lazare has answered my invitation positively. He welcomed the opportunity to lecture in London, as a matter of fact.’

‘Oh, how exciting! When?’ I asked eagerly.

‘Tuesday. The day after tomorrow. He will lecture in our department at three o’clock.’

The professor spoke loudly enough to be heard by people all around the table.

‘A lecture at three o’clock? What’s it about?’ asked several voices.

‘A new light on the Dreyfus affair,’ he replied. ‘Lazare is of course one of the principal proponents of the captain’s innocence. Some new document or other has appeared, supporting the claim, or so he says.’

I took this to be an adroit attempt to turn the conversation
onto the subject of Professor Ralston. It was immediately successful.

‘I heard about some such thing from Ralston before he died,’ remarked a professor of history.

‘Yes, he would have had plenty to say about it,’ said another. ‘He was very active in influencing public opinion in this country against Dreyfus, wasn’t he? I saw an article he wrote somewhere. It was a little shocking, whether the fellow is guilty or not. Even if he did betray his country for money, a professional historian ought to avoid drawing general racial conclusions from it.’