 Cardinality allows us to give a more formal definition for the whole numbers. We'd like to say that one is the cardinality of the set with one element. But we can't. How do we know that the set has one element? We can't define a concept in terms of that same concept. It's like saying blue is the color of blue objects. So, for now we will at least define the whole numbers. The whole numbers are the cardinalities of finite sets. And you might wonder why we specify the sets have to be finite, and we'll get to that. Now, one of the other ideas we have is we could compare whole numbers. We should be able to compare cardinalities. So, suppose A and B are finite sets. We say the cardinality of A is less than the cardinality of B if there is a one-to-one correspondence between A and a proper subset of B. For example, let's find the relationship of the cardinalities of these two sets. So, earlier when we tried to make a one-to-one correspondence, we found the correspondence. And we couldn't do anything with this at the time. But now we can. Since there is a one-to-one correspondence between C and a proper subset, namely this one, of D, then we know the cardinality of C is less than the cardinality of D. And this actually allows us to define numbers. Since the cardinality of A is equal to the cardinality of B whenever there is a one-to-one correspondence between two sets, we could define the whole numbers as follows. One is the cardinality of this set. Two is the cardinality of this set. Three is the cardinality of this set. And we'll stop there. Again, we could define the whole numbers as follows, but there's a reason we don't do this. But we'll get there. So let's prove that this set has cardinality two. Sure, that's easy. Since it has two elements, it has cardinality two. Well, remember, if you're not using a definition, you're probably not doing a proof. And notice that here we have nowhere mentioned what the definition of cardinality is. Let's try that again. Two sets have the same cardinality if there is a one-to-one correspondence between them. That's our definition of cardinality. Now, we did have this definition that two is the cardinality of this set. And so we can look for a one-to-one correspondence between this set and this set. And that's easy enough to find. So there is a one-to-one correspondence between C and the set AB. And since the cardinality of AB is two, then the cardinality of C is equal to two as well. Let's try to prove that one is less than two. And again, we think, sure, that's easy. Since one is smaller than two, then one is less than two. But again, if you're not using a definition, you're probably not doing a proof. So for less than, we had to show that there's a one-to-one correspondence between A and a proper subset of B. And since we want to show that one is less than two, we want to show there's a one-to-one correspondence between the set with cardinality one and a proper subset of the set with cardinality two. So we construct a one-to-one correspondence. And since there is a one-to-one correspondence between the set with cardinality one and a proper subset of the set with cardinality two, then the cardinalities have the less than relationship, which gives us one is less than two. And this finally allows us to properly define the whole numbers. So while we could define the whole numbers by listing specific sets with given cardinalities, that requires us to list a specific set for every cardinality. Instead, we'll begin as follows. Zero is the cardinality of the empty set, so we do have one specific set that has a definite cardinality, but then we'll define the whole numbers as follows. One is the cardinality of any set whose only proper subset is the empty set. And now that we've defined cardinality zero and one, we can define cardinality two as the cardinality of any set whose proper subsets have cardinalities one or zero. And by referring to the previously defined cardinalities, we can continue. Three is the cardinality of any set whose proper subsets have cardinalities two, one or zero, and so on. So earlier we identified that this set has cardinality one. Let's actually prove it using our new definitions. So remember one is the cardinality of any set whose only proper subset is the empty set. So the proper subsets of this are those that contain the elements from the set that are not the entire set. Well, the only set that we can produce that contains elements from this set that doesn't include everything in the set is, but this is the empty set. And so the only proper subset of this is the empty set, and so this set has cardinality one. Again, we claim that this set has cardinality two. Well, let's check it out. The proper subsets are the set containing just A, which we already showed has cardinality one. The set containing just B, which has cardinality one. You should probably prove that. The set that contains nothing, which has cardinality zero. And so the proper subsets of this have cardinality zero or one. And so this set has cardinality two.