There is plenty I could write about countability and uncountability, but much of what I have to say I have said already in written form, and I don’t see much reason to rewrite it. So here’s a link to two articles on the Tricki, which, if you don’t know, is a wiki for mathematical techniques. The Tricki hasn’t taken off, and probably never will, but it’s still got some useful material on it that you might enjoy looking at. The articles in question are one about how to tell almost instantly whether a set is countable and another about how to find neat proofs that sets are countable when they are.



The main additional point I’d like to make about this whole area is that you will do much better if you follow some of the general advice from earlier in this series of posts and work from the formal definitions and basic facts that you have been taught. Perhaps I can make that clearer by spelling out what you shouldn’t do, which is to pay too much attention to the words “countable” and “uncountable”. Let’s face it, you can’t count the natural numbers — you’ll be dead long before you’ve got to . You can’t even put them in a list, since the number of atoms in the universe is only around . And if you imagine some hypothetical world where you live for ever, you’ll never actually finish counting through the natural numbers if you try to do so (unless you find some way of speeding up without limit so that the sum of the times you take converges, but let’s not go there). So if you think of countable as meaning “can be counted”, then you risk confusing yourself — and I know for a fact that many people do end up confusing themselves.

Far better to stick to basic facts and definitions that are stated in precise mathematical language. Here’s a list of them — I may forget one or two important ones but I’ll try not to. You should have all these facts at your fingertips. (If you can prove the ones that aren’t definitions, then so much the better, but knowing the facts is even more important than knowing the proofs, since it is the facts themselves that you will use to go on to prove other things.)

Incidentally, some people use the convention that all finite sets are countable, whereas others use the word “countable” only for infinite sets. I’ll use the convention that finite sets are countable, so if you prefer the other convention then you’ll have to make some small modifications.

1. A set is finite if for some there is a bijection . Otherwise, it is infinite.

2. A set is countable if and only if it is finite or there is a bijection . Otherwise, it is uncountable.

3. Two sets and are said to have the same cardinality if there is a bijection .

4. If and are sets, then has cardinality at most that of if there is an injection .

5. If and are sets and is non-empty, then the following two statements are equivalent.

(i) There is an injection from to .

(ii) There is a surjection from to .

6. Let be an infinite set. The following statements are equivalent.

(i) There is a bijection from to .

(ii) There is an injection from to .

(iii) There is a surjection from to .

[Note that this gives three potential ways of proving that is countable. Although I gave (i) as the definition of countability, it is usually much more convenient to prove (ii).]

7. is uncountable.

8. If is any set, then the power set of has strictly larger cardinality than . (Equivalently, there is no surjection from to the power set of .) In particular, the power set of an infinite set is uncountable.

9. A union of countably many countable sets is countable. More formally, if is a countable set and for each the set is countable, then the union is countable.

10. In particular, a union of countably many finite sets is countable. If you are told to prove that a set is countable, then using this very simple principle usually leads to the shortest proof.

11. If then there is no injection from the set to the set (and hence no surjection from the set to the set ).

[This may look obvious, but it needs a proof. One way to do it is to use the well-ordering principle. Pick a counterexample with minimal. Let be an injection from to . If , then define by taking if and if . This is an injection, which contradicts the minimality of .]

Here are a couple of examples of how to do exercises that involve countability.

1. Prove that if is countable and is an injection, then is countable.

Solution. Since is countable, there is an injection . A composition of injections is an injection, so is an injection from to . Therefore, is countable.

Note how short and clean the above proof is. Note also that what I did not do was say anything about “putting the elements of in a list”.

2. Let be an uncountable set and let be an injection from to another set . Prove that is uncountable.

Solution. Since uncountability is defined negatively, it will be no surprise that we prove this result by looking at the contrapositive. If is countable, then by the previous exercise is countable, contradicting our hypothesis. So is uncountable.

3. Prove that the set of all irrational numbers is uncountable.

Solution 1. There are various ways of doing this. The easiest argument starts from the thought that the reals form a huge set, and to get the irrationals we take away just the rationals, which form a small set. Therefore, the irrationals must form a huge set.

To turn that into a proper proof, we once again prove the contrapositive — for the same reason as we did when solving question 2. If the set of all irrationals is countable, then the reals are the union of two countable sets. Hence, by fact 9 above, the reals are countable. But that contradicts fact 7.

Solution 2. It is tempting to try to use the solution to 2 above. That is, we’d like to find a set that we know is uncountable, and define an injection from that set to the irrationals. The most obvious uncountable set is the set of reals. Can we inject those to the set of irrationals? Hmm, it seems hard, since nothing that’s even slightly continuous has any hope of working. Are there any “less continuous” uncountable sets that we could use? Yes: we could take the set of all 01 sequences. So now we’d like a way of associating an irrational with each 01 sequence. This is fun to do, so here’s a spoiler alert. I’ll leave some space, then I’ll give a solution in just one paragraph, then I’ll leave some more space, and then I’ll present a third solution.

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Bearing in mind that all rational numbers have repeating decimal expansions, we just need some way of associating a non-repeating sequence with each 01 sequence. This can be done in many natural ways, of which one is this. To each sequence associate a decimal between 0 and 1 that is 0 in the th decimal place whenever is not a square, and is either 1 or 2 in the square places according to whether the 01 sequence is 0 or 1. For example, if the 01 sequence begins 00101, then the decimal will begin 0.10010000200000010000000020…

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Solution 3. This is just for people who know a little about continued fractions. If you haven’t, then don’t worry about it — though if you read the beginning of the Wikipedia article on the subject then that will be more than enough to understand what follows.

Continued fractions give a very beautiful bijection between the set of all positive irrational numbers and the set of all infinite sequences of natural numbers. Given a positive irrational number, you just take the terms of its continued-fraction expansion, and given an infinite sequence, you just take the number that has those terms, which must be irrational since all rational numbers have terminating continued-fraction expansions. It is easy to prove that the set of all infinite sequences of natural numbers is uncountable, so we’re done.

Sometimes one is asked to prove that a set is uncountable when you’re not told what is, but just that it has certain properties. This is slightly harder to deal with. I’m not going to work through an example, because I don’t want to spoil what may be a nice examples sheet question from next term. However, here is a technique that can sometimes work very nicely. You define, using information about , a function that takes finite sequences of 0s and 1s to points in , and you do it in such a way that for any infinite sequence, the images of its initial segments form a sequence that you prove converges to something in . (For instance, if the infinite sequence starts 110101… then the sequence of points in starts .) You also do the construction in a way that ensures that no two limits are the same.

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