 So we have the set property called cardinality, which corresponds to what we think about as a number of elements in the set. But it's one thing to have a concept in mind. It's another thing entirely to have a way of talking about it, to have a way of writing about it. And so for that, we'll have to introduce a couple of ideas. First of all, to describe the cardinality of a set, we need to have, first of all, a name for each cardinality. And it's important to understand that these names are completely arbitrary. They are random sounds like, I don't know, Scorch, Gleeb, Nirm... Well, okay, so these are a little bit strange, but they're no stranger than the sounds we do use, 0, 1, 2, 3. The other thing we need is we need a symbol for each named cardinality. And again, these symbols are completely arbitrary, and they're random squiggles like this one, this one, this one. And again, while these look a little bit strange, they are no stranger than the symbols that we do use, this symbol, this symbol, and this symbol. And what is particularly important to remember as we move forward is to keep in mind the difference between what is familiar and what is easy. Counting Scorch, Gleeb, Nirm... is no more difficult than counting 1, 2, 3, 4. They're both completely arbitrary sounds. And again, writing these symbols is no more difficult than writing these symbols. They're both completely arbitrary symbols that have no meaning other than what we assign to them. However, this does raise an important point. Because these cardinality names and these cardinality symbols are completely arbitrary, we have to memorize each and every single one of them. And so the question that we have to answer is how many arbitrary words and arbitrary symbols do you want to have to commit to memory? To answer this question, we'll need to introduce a few terms. We'll say that a system of numeration is base N if we have names for the cardinalities up to and including N. And we have symbols for the cardinalities up to but not including N. And there's a reason for that distinction that we'll see in a moment. For example, a base 4 system has names for the cardinalities up to 4. So we have names for the cardinalities 0, 1, 2, 3, and 4. We also have symbols for these named cardinalities up to but not including the last one. So we have a symbol for 0, we have a symbol for 1, we have a symbol for 2, and we have a symbol for 3. And the reason that we don't include a symbol for the last one is if you look at this point, we have 1, 2, 3, 4 symbols for our base 4 system. Now one important feature to notice is that while we do have a word for 4 things, we don't have a symbol for it. And so the question is how can we write about, how can we talk about the collection that we have shown? We have 4 objects, but how do we write this number? In order to do that, we have to introduce another idea, which is the notion of bundling. To express higher amounts, we introduce the idea of a unit. A unit is either a single object, or it's a set of objects that we treat as a single object. And in base n, each unit is going to be produced by bundling n smaller units together. So for example, suppose we're working in base 3. Our units are going to be a single object. Three of these objects bundled together to form a larger unit. We can then take three of these larger units and bundle them together to form a still larger unit. We can even take three of these units and bundle them together to form a still larger unit, and so on. And now we introduce a central idea of mathematics, which is the following. Arithmetic is bookkeeping. And what this means is that when we do arithmetic, we are ultimately keeping track of how many of which type of unit. And in this case, when we write a number, what we're recording is how many of which type of unit. Now, because we only have symbols for amounts smaller than n, we have to, whenever we have a set of n objects, we have to bundle them together. And now they're treated as a single object, and we do have a symbol for one of something. So our bundling allows us to write down the amounts for any number of objects. We'll start by writing out the amounts dispositionally, where we're going to write out the units explicitly. So again, arithmetic is bookkeeping. We want to write down how many using our abstract symbols and which type using however we're going to represent those units. Let's take a look. So let's see if we can express a number as a number in base four. So in base four, we have symbols for the amounts up to four, but there's more than four objects in this mess. So I can't write down the number of objects using a single symbol. And so what we're going to start doing, because we're working in base four, we're going to start by grouping sets of four. So here's a set of four, and another set of four. Here's another set of four, another, another set of four, and another set of four. And there's no more sets of four single cubes that I can find. But I do have the sets of four of these blocks. So I'll group that set of four blocks into a single larger unit. And at this point, I don't have anything else that I can group as a set of four. So again, arithmetic is bookkeeping, and so we'll write down how many of which type of unit. And so dispositionally, we might write down the amount that we have. We have one big thing. We have two of these medium sized blocks. And we have three of these small cubes left over. And one of the advantages to dispositional notation is that as long as we write down the units, the order doesn't matter. One big thing, two medium, three small is the same as two medium, one big, three small. And for that matter, it's the same as three small, two medium, one big. Dispositional notation order doesn't make a difference because arithmetic is bookkeeping, and we are in every case giving how many of which type of unit.