 So, let's talk about counting. Now, informally again, the cardinality of a set is the number of things in the set, and so cardinality is a set property. However, our ability to talk about cardinality, and later on to actually do something with cardinality, really depends on our ability to create new words. If you can't talk about it, you can't really communicate about it, and again, how you speak influences how you think. So, there's a limited number of ways we can put sounds together, so we generally proceed as follows. So, I create a name for the cardinality of the empty set, and we've actually done that already. We call the cardinality of the empty set zero, and then I look at the cardinality of the set with the next larger size. So, I have the empty set, and I'm going to think about the set that's just larger in size than the empty set, and I need a name for the cardinality of this set. Thinking, thinking, thinking. I'll call that cardinality of that set, global. Well, maybe let's call it one. And then, I now have two sets, and I can think about, well, what's the cardinality of the next larger set? And so, again, I'm going to repeat this process until I either run out of creativity. Well, that's actually unlikely. Human beings are fairly creative, but the problem we do run into is that you can name all these things that you want to, but sooner or later, you're going to have to remember all of them. So, at some point, we have as many names as we care to remember. There is a last named cardinality, and the base of our system is going to be the highest named cardinality. Now, there's a little bit of a metaphysical weirdness in here that we'll get to in a moment, but for right now, we can say that the highest cardinality we care to name is the base of our system. So, for example, let's say that this is as far as we care to go. This is the largest set where we care to name the cardinality of this set, and so we want to construct a naming system and find the base. So, we want to find all sets that are smaller than this one and give names to all of the sets, including this set in front of us. So, let's see. So, I'm going to start with the empty set. So, here it is. I will draw very carefully the empty set here. There it is. And the cardinality of this, I need a name for it, and we've already given that a name. We'll call it zero. I'll draw a set with the next bigger set, and this is the set that contains this only. And so, I need a name for this, and again, we've used the name one for this set, and a look at the next larger set. And so, now I have these things here, and I need a name for the cardinality of that set. Thinking, thinking, thinking. Call it two. Next larger set, cardinality. We need a name for it. Call it three. And finally, this set, which is the largest set whose cardinality we care to name. We're going to call that four. And the base of our system is going to correspond to the highest named cardinality. In this case, our cardinality of this, the highest name we care to make, is four. So, our system is going to be base four. And that's great if this is the largest thing you ever have to deal with. Unfortunately, we probably have to deal with larger things. So, what do we do at that point? In order to go further, what we have to do is you have to introduce a concept, which we call the concept of a unit. And in base n, whatever n happens to be, a unit is either going to be a single thing or it's going to be a set of n things which I'm going to treat as a single thing. And the useful thing to keep in mind here is that when we talk about an object, these objects might be other units. So, a unit is either a single object or it's a set of n objects, and the objects themselves might be other units. And the advantage of this is we can count to higher cardinalities and the only names we have to come up with and remember are the names of the units. For example, let's continue to count in base four and we'll show, we won't worry about naming them because that takes a bit of creativity, we'll name them later on, but we'll show at least the higher units in base four. So, again, our first two units, we have a single object and a set of one, two, three, four of these objects. So, here's our first unit, our single object, and here's our next unit, which looks something like that. Now, it's convenient to have a name for this unit and in a fit of creativity, we'll say that this unit, we're going to call it a, let's call it a, we'll call it a four and now I can start to look at sets of fours. So, here's two fours. So, there's a four back here, there's another four up front, there's two fours, three fours, four fours and now I have four of something and I have four units. This is a new unit and it's a collection of four fours and I need a name, so let's call it a four four. Perfectly natural thing to do if we weren't as, if we weren't as educated and hadn't had as many years of schooling as we do, we might call it something else, but anybody introduced to this idea for the first time is going to call it a four four because that's what it is. And it's a unit, so now I can count by four fours. So, well, there's one, there's two, there's three, there's four and here, again, I have four of these four fours and so this is also a next higher unit. And again, we need a name for this and again, with our years of schooling, we can come up with a name for this fairly easily, but again, if I were being introduced to this concept for the first time, I might continue a winning strategy and call this a four four four and so I have my units, a one, a four, a four four and a four four four and except for the peculiarity of the names, these are perfectly legitimate units and perfectly legitimate names for those units in our base four. What do we do with them? Well, let's see how we can count in the next section.