 Developing the theory of the real numbers usually begins with piano axioms, first expressed in 1899 by the Italian mathematician Giuseppe Piano. There's a couple of different versions of the axioms, but one version is the following. First, there is some set N, which we call the set of natural numbers. Now, remember one of the questions that you should always ask when you're presented with the set is, is there anything at all in the set? And so the first axiom is, there is an element zero in the set of natural numbers. This inclusion of zero as a natural number is due to Piano. The more important feature here is that because we've identified there is something in the set, this tells us that our set of natural numbers is a non-empty set. Another important idea to keep in mind is that zero is just a symbol with no special properties. And we could have said that this is an element of the set of natural numbers. It doesn't matter what symbol we use. The second axiom, if N is a natural number, then there is a unique N star, which we call the successor of N, which is also in the set of natural numbers. So N in the set of natural numbers, meaning that N star is also in the set of natural numbers, means that for any element of the set of natural numbers, its successor is also in the set. There's no requirement on what that successor has to be yet. And so we might say that spade, successor, is club. And if spade is a natural number, then club is also a natural number. The third axiom, if M star equals N star, then M is equal to N. And what this means is that if two numbers have the same successors, they have to be the same. So spade star equals club and diamond star equals club as well, we can conclude that spade is equal to diamond. Now, while every number has a successor by the second axiom, there is no N for which zero is N star. And this means that zero is not the successor of any number, and we can think about it as the first natural number. More generally, while at the beginning there were no special properties of zero, now we've identified a special property. And so zero is a special number. Now, the fifth of the pano axioms is a complicated one. Suppose zero is in some set, and whenever M is in the set, it's also true that the successor of M is in that set. Then the set of natural numbers is a subset of S. This means that if S contains zero, and it contains a successor of every natural number it contains, then S contains all the natural numbers. And this is known as the principle of mathematical induction. Now, remember the piano axiom specifies that every natural number has a successor, but we were unspecific about what that successor was. And so then piano gives the following definitions. The successor of zero will call one, the successor of one will call two, the successor of two will call three, and so on. It's important to remember that the symbols zero, one, two, and so on have no meaning beyond this definition. To emphasize that point, while we'll use the piano axioms to create a theory of the real numbers, it's important to understand that they can be applied to any ordered set. For example, let pi be the set of prime numbers, let's interpret the first four piano axioms, and then find the successor of five. So let's pull in the piano axioms, and the first one is easy to translate. There is a set pi of prime numbers. Now, the first of the piano axioms says there is an element zero in our set of natural numbers. Now, zero is not in the set of prime numbers, but remember it's a symbol with no particular meaning, and any other symbol can replace it. We'll figure out which symbol later. So the second of our piano axioms says that we have successors of each of our elements. So let's think about that. If P is in the set of prime numbers, then P star should also be in the set. And so since P star is a successor of P, and P is a prime number, we might view this as the next prime after P. And the set of primes is infinite, so for any prime we might view P star as the next higher prime. And so this gives us a way to restate that second axiom for any P in pi. The successor of P is also in pi, where the successor is the next higher prime number. Now, that gives us insight on what this symbol zero should be. If we view P star as the next higher prime, and we want zero not to be P star for any P, this means that our symbol zero should be the lowest prime number. So instead of the symbol zero, we'll use the symbol two. And so that completes our restatement of the first axiom. And because two is the first higher prime number, this also allows us to restate the fourth axiom. Two is not the successor of any prime number. The third piano axiom doesn't have to change. Nor does the fifth. And finally, how about five star? So remember star is the next higher prime number, and since five star is the next higher prime, then five star is equal to seven.