 So how can we define multiplication using the k-on-apostulates? And so our definition is this. We're going to define n times 1 to be n itself, and then n times the successor of k is n times k plus n. Now let's prove an important theorem for any natural number. Multiplying by 0 gives you 0. So you know we're going to need the definition. And from our definition, we know that n times 1 is equal to n. Now we'd like to say something about 0, and the thing we might remember is that 1 is the successor of 0. So n times 1 is the same thing as n times 0 star, but our definition of multiplication says that that's n times 0 plus n. And since these two are the same thing, we know that n is equal to n times 0 plus n. Our definition of addition tells us that n and n plus 0 are the same thing. We went to a lot of trouble to prove commutativity, so n plus 0 is the same thing as 0 plus n, and we subtract n from both sides. Wait, we can't do that because we haven't defined subtraction, and we haven't proved that you can subtract the same thing from both sides. On the other hand, we have proven cancellation that if we have the same thing on both sides, we can just get rid of it, and so we get 0 is equal to n times 0. And from this point, we can prove all the usual rules of arithmetic of addition and multiplication. Again, we're not going to do all your homework for you. We'll leave most of these proofs as exercises. All right, we'll do 1. Let's prove that n times 1 is equal to 1 times n, the commutativity of multiplication by 1. So we notice that this is the same as proving that 2 times 1 equals 1 times 2, 3 times 1 equals 1 times 3, 4 times 1 equals 1 times 4, and so on, so an induction proof seems appropriate. So our base step, we want to prove that 1 times 1 is equal to 1 times 1. Okay, sometimes proving the base step is pretty easy. Now, our induction step, suppose our statement is true for n, we would like to show that it's true for n star. So let's consider n star times 1, our definition of multiplication tells us that that is n star. 1 times n star, our definition of multiplication tells us that's 1 times n plus 1. Now, we assume that 1 commutes with n, so that's n times 1 plus 1. Our definition says that n times 1 is n, and we also proved a while back that n plus 1 is the same as n star, and so n star times 1 is n star, and 1 times n star is n star. And so if n times 1 is 1 times n, then the same thing is true for the successor. Now, the piano axioms allow us to prove all the standard theorems about the arithmetic of the natural numbers, the additive identity 0, straight out of the definition of addition, commutativity and associativity of addition, the multiplicative identity 1 by the definition of multiplication, commutativity and associativity of multiplication, and the distributive property of multiplication over addition. There's just one troublesome little problem. The piano axioms aren't specific to the natural numbers. As we saw, they could be applied to any set with the concept of first and successor, and so the question you might want to ask is, could we develop a theory of the natural numbers only? And in particular, is there some way we can identify what we mean when we say 0 or 1 or 2 or any other natural number? And we'll take a look at that next.