Why "well-defined" is the whole difference between a set and a heap

0:00Here is a collection. The five greatest musicians who have ever lived. Pause for a moment and actually name them. Say them out loud, or write them down. Done? Good. Because I have my five as well, and I will bet you almost anything that our lists do not match. Now here is a second collection. The odd numbers below ten. One, three, five, seven, nine. You got that. I got that. Everybody gets that.

0:31Two collections, two very different experiences. And the difference between them is not a small matter of taste. It is the difference between something mathematics can work with and something it cannot touch at all. So let us put six collections on the table and look at them properly. The odd numbers below ten: one, three, five, seven, nine. The rivers of Africa. The vowels of the English alphabet: a, e, i, o, u.

1:03Triangles of various kinds. The prime factors of two hundred and ten, which are two, three, five and seven. And the solutions of the equation x squared minus five x plus six equals zero, which are two and three. Look at how different those are from each other. Two of them are letters and rivers, not numbers at all. One is endless, if you are willing to count triangles. And yet something is the same about all six.

1:36Here is the thing they share, and it is worth saying slowly. Pick any object at all. Anything. Then ask: is it in this collection? For every one of those six, that question has exactly one right answer. Not an answer that most people would agree on. Not an answer that experts would settle. Exactly one right answer. And crucially, that answer does not depend on who is asking. You could be in a different country, in a different century, in a very bad mood, and you would still get the same verdict.

2:13That is not a description of the collection. It is a test the collection carries with it. Let us run the test on one that is not made of numbers. The rivers of Africa. I hand you the Nile. Is it in? Yes. It runs through Africa, so it passes. Now I hand you the Amazon. Is it in? No. It is a river, and a very large one, but it is on the wrong continent.

2:43Notice what just happened, because it is easy to miss. I never showed you the list. I could not have. I do not know every river in Africa and neither do you. But we settled both questions without hesitating, and we would have agreed on any other river you handed me. The test works. The list was never needed. Now the collection that fails. The five most renowned mathematicians in history.

3:13I ask two careful people to write the list. Both are serious, both know the subject. The first ranks them by how much their work is still used today. The second ranks them by how far ahead of their own time they were. Two reasonable standards. Two different lists. And here is the part that matters: there is now a name that the first person put in and the second person left out.

3:42That name is in the collection and out of the collection at the same time. Not because anyone made a mistake. Both people applied the description exactly as it was given. The description simply never said which standard to use, so it handed the decision to whoever was holding it. Now, the wrong lesson to take from that is: avoid superlatives. Do not say best, do not say greatest, and you will be fine. That is not it.

4:15Take the eleven best football players in the world. That fails, for exactly the reason we just saw. Now change six words. The eleven players who have scored the most goals this season. Still a superlative. Still picking a top eleven. But now hand it to those same two people. They go to the same figures, they count, and they come back with the same eleven. There is nothing left for their taste to bite on.

4:45So it was never the word best that broke the first one. It was the missing standard. Supply the standard and the superlative becomes perfectly respectable. Which gives us the definition, and now the words in it should feel earned. A set is a well-defined collection of objects. Well-defined is doing all the work in that sentence. It means the collection comes with a test, and the test returns the same verdict no matter who runs it.

5:19That is it. That is the whole entry requirement. And notice what it does not require. It does not require that you can write the members down. It does not require that there are finitely many. It does not require that they are numbers. A collection that fails this is not a badly behaved set. It is not a set at all, and nothing that follows will apply to it.

5:47A little housekeeping now, and it is not decoration. Sets get capital letters. A, B, V, X. The things inside them get small letters. a, b, v, x. So when you see a small letter next to a capital, you already know which is the container and which is the contents. One more piece of housekeeping. You will hear three different words for the things inside a set. Element. Member. Object.

6:16They mean the same thing. Not almost the same thing, not subtly different things. The same thing. If you meet all three in one paragraph, nothing has changed; the writer just varied the word. Now the symbol, and it has exactly one job. It records the answer the test gave. Let V be the vowels of the English alphabet. Then a belongs to V. We write a, belongs to, V.

6:46And b does not belong to V, so we strike the symbol through. That is the entire meaning. The symbol is not a claim, not an opinion, not a definition. It is a receipt. The test was run, and this is what came back. Which is why it can only ever be used on a genuine set. Run it on a collection with no test and there is no answer for it to record.

7:14Here is the example that catches the most people, so watch it carefully. Let P be the prime factors of thirty. Thirty is two times three times five, so P holds two, three and five. Is three in P? Yes. Three is prime and it divides thirty. Now: is fifteen in P? Fifteen divides thirty exactly. Two fifteens make thirty. So it feels like it should be in. But it is not, and the reason is that the test does not say factor. It says prime factor.

7:50Fifteen is three times five, so it is not prime, and it fails the test at the first word. Thirty has eight factors altogether, and only three of them are prime. Read the test, not the words around it. That single habit will save you more marks than almost anything else here. Some sets come up so often that they get permanent names, and these will follow you for years.

8:19N is the natural numbers: one, two, three, and onwards. Z is the integers, which adds zero and the negatives. Q is the rationals, every number you can write as one whole number over another. R is the reals, the whole number line. Then three more, each with a small plus. Z plus, Q plus, R plus. Each of those means: that same system, but only the positive members. Seven symbols in total. And you may have noticed there is no N with a plus.

8:55There is no need for one. The naturals have no negative members to throw out, so the positive-only version would be the same set again. Let us finish the sorting job, because two wrong ideas hide in it. The months whose name begins with J. That is a set: January, June, July. Three of them, settled. All the boys in your class. A set. You could take a register. All the natural numbers below one hundred. A set, and a large one.

9:30All the even integers. A set, and an endless one. Every novel written by one named author. A set. The author is fixed, so the list is fixed. All the tracks on one particular album. A set. The album is a fixed thing, so what is on it is settled. And then the three that fail. The ten most talented writers. The eleven best footballers. The world's most dangerous animals.

10:00Every one of those three is a ranking with no stated standard, and every other one on the list carries a real test. So being large did not disqualify anything. Being endless did not disqualify anything. Only the missing test did. One last thing, and it is the honest part. Well-defined is a working idea, not a finished foundation, and mathematicians found that out the hard way. If any description with a clear test gives you a set, try this description.

10:37The collection of all collections that are not members of themselves. That has a test. It looks perfectly respectable. So ask whether that collection belongs to itself. Suppose it does. Then by its own rule, it must not. Suppose it does not. Then by that same rule, it must. There is no consistent answer, and that was discovered more than a century ago. It did not destroy the subject. It showed that the definition we have been using needs more care at the edges than it looks like it does.

11:15But for everything you will meet, the working idea holds, and it is enough. So here is one to argue about. Is the collection of all interesting numbers a set? Because the moment you try to find the smallest uninteresting number, something odd starts to happen. Tell me in the comments.