In the previous lesson we used our new weapons for counting and applied them to “counting” infinite sets. Namely, we saw that we can “double” the number of elements in a set with infinity type 1 elements, and get a new set which also has infinity type 1 elements. We also saw that we could add any finite number of elements to a set with infinity type 1 elements, and still have a set with infinity type 1 elements. Recall that whether or not a set “still has infinity type 1 elements” is determined solely by whether or not we can define a bijective function from to the set.
But last lesson was just a warm-up lesson for the next couple of lessons. The statements of the next two lessons, and their proofs, will take some more creativity and perhaps a bit more thought. That said, I can promise two things. One is that it won’t be that bad. And the second is that the results will make the process entirely worth it (as is usually the case with math).
So what is it that we want to prove in this lesson and the next one? We’re going to show that the set of all fractions also only has “infinity type 1” elements in it. This is a truly remarkable result, and in order to fully appreciate it we need to first remind ourselves about fractions and the nature of their infinity. This is the subject of this lesson, saving the actual proof for the next one.
Recall that fractions are nothing but things like (“one-half”, or “1 divided by two”), or or . Namely, “” is a symbol which represents the abstract idea of the number that, when multiplied by 2, gives 1. In the same way, “” is a symbol which represents the abstract idea of the number that, when multiplied by 4, gives 3. Accordingly, we can think of fractions as nothing but a pair of whole numbers: the number that is “on top” and the number that is “on the bottom”. Thus, in the fraction 3 is the number “on top” and 4 is the number “on the bottom”. Given a fraction, if we multiply it by the number that is “on the bottom”, then we get the number that is “on top”. I.e., and This might all be familiar, but if it’s not I hope this suffices as a reminder. Lastly, recall that we usually call the number “on top” the numerator, and the number “on the bottom” the denominator. Fractions are not scary at all when you recall that a quarter of a basketball game is just of a whole game, or that a nickel is just of a whole dollar, or that a cup can be full.
The last thing we need to recall is that there is a notion of equality between two fractions that might at first sight look different. Namely, we have that and are the same. Also, and are the same fraction, as is and The general idea is that if we have one fraction and we obtain another fraction by multiplying “the top” and “the bottom” by the same number, then the resulting fraction is actually the same as the one we started with. Thus, when I define the set , I’m implying that and are the same element in FRAC.
Now that the crash course on fractions is over, let’s see what makes the fractions seemingly “more infinite” than There is a very important fact about fractions that has no analogue in the set Namely, between any two distinct fractions, there are infinitely many more fractions! Seeing why this is not the case in is easy. Suppose you gave me two distinct elements in Call them M and N. One of these two numbers will be bigger than the other (since they’re different), so let’s suppose N is bigger than M. Then clearly there are only finitely many elements between M and N, because all we need to do is count off until we hit N. Since both M and N are finite numbers, we’ll only have to count off finitely many elements.
Let’s take a classic example now to see why there are infinitely many elements between any two distinct fractions. Let’s ask the question: how many fractions are there between and 1 (Recall that whole numbers are fractions whose “bottom” number is just 1, i.e., )? Well, what we’re going to do is essentially just take the fraction “halfway” between and 1, and then the fraction halfway between that fraction and 1, and then the fraction halfway between that fraction and 1, and so on. We can do this infinitely many times without ever reaching 1, thus finding infinitely many fractions between and 1. Thus, is between and 1, is between and 1, is between and 1, is between and 1. We can keep doing this forever, thus uncovering infinitely many elements between and 1 in FRAC.
The point is that we can make this argument between any two fractions. All we do is step halfway from one to the other, then halfway again, then halfway again, and so on. Since each of these “halfway points” will be new fractions in their own right, we’ll uncover infinitely many fractions between any two distinct fractions.
Now here’s the rub. Suppose the enemy hands me two distinct fractions. From the above considerations, we know that we can find infinitely many fractions between these two fractions. But in fact, we can do better! For between any two of these fractions that we’ve now found, there is yet another infinity of fractions! In other words, every time we take a step halfway from one fraction to another, we can find another infinity of fractions between the two we just “stepped between”! Moreover, if we then dive into that infinity of fractions, we’ll keep uncovering new infinite sets of fractions.
Let us go back to our example to see just how infinite the set of fractions are. We know that there are infinitely many fractions between and 1. One of the fractions between them is But now we automatically know that there are infinitely many elements between and , because we can just do the same “half-way” argument. One fraction between and is . Again, we now know that there are infinitely many elements between and ! One such fraction is . Then BOOM! There’s another infinity of fractions between and ! In this way, infinities just keep “popping out” everywhere we turn. Namely, we just keep picking different fractions, no matter how close they are, and there will be infinitely many fractions between them. ALWAYS!
This sort of infinity certainly has a different quality about it than the infinity type 1 of Namely, just kind of “goes in one direction”, whereas we can “zoom in” on the fractions infinitely far, in infinitely many places, infinitely many times! This is what I mean by “infinity times infinity”. Between any two of the infinitely many elements in FRAC, there are infinitely many elements, and between any two of those elements, there are infinitely many elements, and between any two of those elements, there are infinitely many elements, and on and on and on!
Believe it or not, we actually still can define a bijective function from to FRAC. To see that we can define an injective function is easy, since every whole number is itself a fraction. Thus we could define an injective function just by sending 1 to 1, and 2 to 2, and 3 to 3, and so on. But this function certainly isn’t surjective. In fact, it might seem pretty unlikely that we can “hit” all of the fractions just with , but we actually can! It will take some work, and this will be the subject of the next lesson. Don’t lose faith though, pretty soon we will see a whole new kind of infinity!