In the previous lesson we saw how we could use our notion of bijectivity to give a rigorous meaning to our intuitive idea of counting the elements in a set. Namely, if we had a set A and there was some N (a positive whole number) such that the set could be mapped bijectively into our set A, then we define our set to have N elements. It is important to note the flow of logic in this construction. We make no reference to “how many” elements are in A before we start looking for bijections. Instead, we simply consider the sets , and so on, until we find one that lets us define a bijective function from it to our set. Note that if we find an N that works, then there will be several different bijections from to our set, which is okay. All that matters is that there is at least one! The last couple of lessons showed that if we can find one such set, then no other set of that form will do. In particular, if we find such an N (such that works), then we know that this is the only N that will work. This all has to do with the nature of bijections, and if the reader is rusty on this then he or she should consult the last two lessons.
This all seems like a stupid amount of work and abstraction in order just to do something as trivial as counting. However, we will see in this lesson that this way of doing things is actually immensely powerful, and it will allow us to make conclusions about “infinity” that we never could have made without these tools at our disposal.
One very important subtlety that the astute reader may have picked up on is that the above construction simply doesn’t work if the set A that we were given has infinitely many elements. Why is this? Well, if we’re given a set with infinitely many numbers, then by definition of infinity, there will be no N such that we can put the set into bijection with A. For if A has infinitely many elements, then clearly can’t be put into bijection with it, nor can , nor can , nor for any N!
Let us take a trivial example of an infinite set: Let . I.e., let where now the “…” means that this set “goes on forever”, including only the positive whole numbers. (Don’t be confused by the fact that we’re using the positive whole numbers to define our counting. Remember that if we have , then B can obviously be put into bijection with , and so B has 5 elements. All I’m saying is that there’s no reason why we can’t apply our counting rules to the sets themselves.) Clearly, there is no N such that the set can be put into bijection with A, simply because “stops” somewhere (at N), whereas A doesn’t.
Well that’s a bummer. Is it the case then that there are sets that we simply can’t count? I.e., are there sets that are immune to the high-powered weapon of counting that we just developed? If the answer were yes I probably wouldn’t have written this lesson, so I’m sure you can guess what the answer is. What this problem is telling us is that we simply need to make our weapons more powerful.
And how do we do that? Well, let us extend our number system to account for these problematic cases. Moreover, let us do so in the most obvious way possible. While what follows may seem like a “stop-gap” of sorts, the results that we’ll derive in this and the coming couple of lessons will show that there is in fact some kind of deep “truth” to all that we’re developing.
So how do we extend the numbers? Easy! As before, we say that if a set can be put into bijection with the set for some N, then that set has N elements. Now all we need to do is say that if a set can be put into bijection with the set (note that this set doesn’t stop), then that set has “infinity type 1” elements. In order to avoid extremely awkward language, let’s make a quick and easy definition:
Definition 9.1 If for some N, a set A can be put into bijection with the set , then we say that A has cardinality N. Moreover, if a set A can be put into bijection with the set , then we say that A has cardinality infinity type 1. //
Thus, the cardinality of a set is just the number of elements in that set, where “infinity type 1” is just viewed as some new infinite number. Recall from my note on notation that “3” is just a symbol for the abstract idea of “the number three”, which in our case translates to the abstract idea of “a set’s ability to be put into bijection with the set “. In the same exact way, if we view “infinity type 1” simply as a symbol (just like “3” or “16” or “1,245,695”), then what that symbol stands for is the abstract idea of a set’s ability to be put into bijection with .
Now, the astute reader is likely wondering why I’m calling this new number “infinity type 1”, as opposed to just “infinity”. Is there an “infinity type 2”? What about type 3? Surely there can’t be more than one type of infinity, right? I recall being on the playground as a younger child and arguing with a buddy about who a girl liked more—me or him. I argued that she liked me more, and he argued that she liked him more times 2. After some quick analysis of the data, I concluded that she in fact liked me more times a million, but was immediately dealt the death blow when I came to learn that she in fact liked him more “times infinity”. That was it, I was toast. My first grade crush was forever lost, simply because there was nothing I could say that could bring my “likeability” up over infinity. The simple reason for this is that infinity plus anything is still infinity, and infinity times anything is still infinity.
What a poor, unfortunate first grader I was, for I simply didn’t know at the time that I was not in fact out of the fight for my beloved! While it is of course true that anything times infinity is still infinity, it is not true that there is nothing greater than infinity. In fact, what I should have asked my competitor was which infinity he was talking about, because no matter which infinity he was talking about, I’d be able to find one that is greater! Unfortunately, I still wouldn’t have necessarily won (but rather would have just stayed in the fight longer), because this back-and-forth of greater and greater infinities in fact can go on forever! In fact, there is an infinitely high tower of infinities, reaching off into the abyss, of ever-increasing infinities!!
This might all sound more like fantasy than mathematics, but throughout the next 4 lessons we’ll make all of this completely clear and rigorous, thus equipping you for your next verbal battle for love. In the next lesson, though, we’ll take on the much more modest task of showing that infinity plus anything is indeed “the same” infinity, and that “infinity times 2” is also the same infinity. After that we’ll go on to show that even “infinity times infinity” is still “the same” infinity. We will continue to press on though, against all odds, and actually uncover this infinite tower of new and ever-increasing infinities. Let’s go!