A set with nothing in it, and sets you cannot finish listing

0:00Two collections of students, and the difference between them is not a difference of size. First: the students in one particular class, at one particular school, today. You could settle that one by walking in and counting heads. Nobody has. The number still exists. Second: the students who are in Year Ten and Year Eleven at the same moment. Not the ones who moved up last year - the ones in both, right now.

0:30Go and look. You will not find one, and not because the school is small - you will not find one because being in Year Ten means not being in Year Eleven. So the first collection has a count that nobody has bothered to take. The second has no members at all. Both are sets. This video is about deciding which is which without ever seeing a list. A set with no members at all gets a name of its own.

1:02It is called the empty set, and also the null set, or the void set. And why THE empty set, rather than AN empty set? Because two sets are the same set exactly when they have the same members. So take any two collections that have nothing in them. Do they have the same members? Yes - neither of them has any. So they are the same set. Not two similar sets: one set, written twice.

1:31That is not a convention anybody chose. It falls out of what equality of sets already means. There is a single symbol for it, phi, and there is a pair of braces with nothing between them. Those mean the same thing. The empty braces are the more honest, because they show you the claim: here is a set, and here is what is inside it, which is nothing. Now the trap.

1:59Braces with phi inside them. That is not the empty set. It is a set with one member, and the member happens to be the empty set. A box with an empty box inside it is not an empty box. Count them and the difference is immediate. The empty set has no members. This one has one. Four conditions now, each of which admits nobody, and each for a completely different reason.

2:30Work them. Do not recognise them. First. The natural numbers strictly between one and two. That is an ordinary sentence. Nothing in it looks broken. But the natural numbers go one, then two, then three. They step. There is no natural number between one and two, because the step from one lands on two and does not pause on the way. Now notice what that argument actually depended on. Ask the same question about fractions and the answer changes completely. Three halves is between one and two. So is five quarters, and endlessly many others.

3:12The emptiness was never in the words alone. It was in the words together with the numbers you were allowed to answer from. Second. The rational numbers whose square is two. A rational number is one whole number over another. So is there one that squares to exactly two? Search, and you will not find it. But not finding something is weak evidence, so here is the argument. Suppose there were such a fraction, written in lowest terms.

3:45Then the top squared is twice the bottom squared. So the top squared is even. And a number whose square is even is itself even. So the top is even. Put that back in and the bottom squared comes out even too, so the bottom is even as well. Both even. But we said lowest terms, and lowest terms means they share no factor. So there was no such fraction to begin with. Not undiscovered. Non-existent.

4:16Third. The even prime numbers above two. Every even number above two has two as a divisor - not one, and not the number itself. So it is composite. Two itself is prime, and two is even, and two is the only number that is both. Above it, nothing. Fourth. The odd numbers whose square is four. Solve it rather than staring at it. What squares to four? Two, and minus two. That is the complete list.

4:49Two is even. Minus two is even. So the condition is asking for something odd out of a collection that contains nothing odd, and it comes back with nobody. Four conditions: a number system that steps, a square root that is not a fraction, a divisibility fact, and a solved equation. Not one of them was empty because it was badly written. Each was settled by an argument, and a different argument each time.

5:22Now, counting. When a set has a definite number of members, we write n of S for that number. Take the set holding one, two, three, four and five. n of that set is five. Take the set holding a, b, c, d, e and g. Look at it again before you answer. n of that set is six. And most people get six by thinking the first six letters of the alphabet, which is the wrong route to the right answer.

5:52It skips f. The letter f is simply not in there. The count survives the mistake, which is why the mistake survives with it. Here is a third set. Every person alive on Earth right now. Nobody knows that number. It changes while you are saying the sentence. So is that set infinite? No. It is finite. There is a definite number of people alive at this instant. Nobody has it, nobody can get it, and it exists.

6:26Finite does not mean counted. It does not mean countable in an afternoon. It means there is a number. Unknown is not infinite. That is the most useful sentence in this topic. People merge them constantly, because too many to count and never ends feel like the same complaint. Which brings us to the definition, and one word in it is doing deliberate work. A set is finite if it is empty, or if it has a definite number of members.

7:00Empty. Right there, in the opening clause. The empty set is not a third case sitting outside the split. It is filed under finite, on purpose. And you can see why. n of the empty set is zero, and zero is a perfectly definite number. Leave that word out and a completely ordinary set would have had nowhere to live. Infinite is then defined by exclusion. A set is infinite when it is not finite, and that is the entire definition.

7:35Notice what it does not say. It does not say very many. It does not mention size at all. All it claims is that no number counts these. Which is why the people alive on Earth are finite though nobody can count them, and the multiples of five are infinite though they are easy to describe. Now, writing an endless set down. The natural numbers: one, two, three, and dots. The odd numbers: one, three, five, and dots. The integers, running out in both directions.

8:12Every one is written with three dots, and every one is legitimate - because the dots are a promise, and here the promise is kept. The promise is this: you can work out what comes next. One, two, three, and the next one is four. One, three, five, and the next one is seven. There is a rule sitting underneath the first few members. It regenerates them, and then it keeps going.

8:39Dots are not shorthand for and so on. They are a claim that the next member is predictable by the reader. And here is the set where that claim collapses. The real numbers between zero and one. Try to write it with dots. To start, you need a first member. Name one. Whatever you named, take half of it. Smaller than yours, still above zero, still in the set. So yours was not first. And neither is the new one, by the same move.

9:13There genuinely is no next one. Between any two of these numbers there is always another, so no member ever sits immediately after another member. Which means there is nothing for the dots to promise. Hand a reader the first few and they cannot produce the next one, because there is none. This set is infinite, and it cannot be written as a list at all - not a longer list, not a cleverer pattern.

9:42Endlessness comes in more than one kind, and the dots only cover one of them. Five conditions on the natural numbers. Finite or infinite? Work each down to its members first, then judge. x minus one, times x minus two, equals zero. That gives one and two. Two members. Finite. x squared equals four. Over the naturals, that gives two, and only two, because minus two is not a natural number. One member. Finite.

10:17Two x minus one equals zero. Solve it, and x has to be a half. A half is not a natural number, so nothing at all satisfies this one. It is the empty set. And by the definition we just read, the empty set is finite. So this one is finite too. That third case is why the empty set had to be named before this question could be asked.

10:45x is prime. The primes never run out, so that one is infinite. x is odd. Infinite. Three finite, two infinite. And two of the three finite ones had to be solved before they could be counted. Last, a sort, with two traps buried in it. The months of a year. Twelve of them. Finite. The natural numbers from one to a hundred. A hundred of them. Finite. The positive integers above a hundred. Infinite.

11:18The prime numbers below ninety-nine. Finite: twenty-five of them, largest ninety-seven. The letters of the English alphabet. Twenty-six. Finite. The multiples of five. Infinite. Now the two traps. The even prime numbers. That sounds like it should be empty, and it is not. It is the set containing two. One member, not none. And the odd numbers divisible by two. That one really is empty, because odd already means not divisible by two.

11:55They read like the same kind of sentence. One admits somebody, one admits nobody, and the only way to tell is to work it. One more, and it needs a word said out loud: the points shared by two DISTINCT parallel lines. Distinct matters. Parallel and distinct means they never meet, so there are no shared points at all and the set is empty. So: two verdicts, and one method underneath both.

12:26Empty or not empty, finite or infinite: neither is answered by looking at a list, because for some of these there is no list to look at. Both are answered by reasoning with the condition, which is the whole reason a set is worth describing by a condition. A condition that admits nobody is not a broken condition. It describes a perfectly good set, the one with nothing in it.

12:56And a condition that never runs out is not describing a very large set. It is describing one that no number counts. Here is one to try for yourself. Write a condition on the natural numbers whose set is empty, then prove that it is empty, rather than just failing to find a member. Those are different achievements, and only one of them is mathematics. Tell me what you came up with.