The natural numbers. The set of natural numbers is the set usually denoted by an N with an extra stem which contains the numbers zero, one, two, three and so on. Now, there are two ways of viewing this set. The first one is the view of the German mathematical mathematician Kronecker who lived in the 19th century, 1823 till 1891. And his view on the natural numbers was quite simple, he said, "The natural numbers have been created by God. All others, all other numbers that is, are the handiwork of man. And for someone who was only using numbers as a tool, as I said, this is quite acceptable. Natural numbers are there, we've learnt them at primary school. You can add them, you can multiply them, you know everything what to do with them. But of course, for mathematician this is not quite satisfactory. So, I would like to give you an insight in how the natural numbers are defined within mathematics, not because you will need this but it shows you the kind of details mathematicians are obsessed about. The existence of the natural numbers is postulated by a number of axioms, the so-called Peano axioms for the natural numbers. Giuseppe Peano was an Italian mathematician who lived also in the 19th century mostly from 1858 till 1932 and this was probably his most important contribution to mathematics. There are five axioms and they are quite simple actually. The first one says zero is a natural number. Actually he said that one was a natural number but later on this was corrected to zero because it's too many people more logical to start with that. The second axiom was, every natural number N has a successor and prime that is also a natural number. This just expresses that we can count. There is zero and there is the next number one and there is the next number which we call two and there is the next number which we call three et cetera. But of course, we have to make sure that by discounting after four that we are not returning to zero. So that's the next axiom is about zero is not the successor of any natural number. So, we never can return to zero by counting up. Then, natural numbers whose successes are equal, are themselves equal. In other words, if we've got two natural numbers and we want to see whether they are equal we can say, okay they're equal if their predecessors are equal and good those are equal those predecessors equal and then you go down until there's no more predecessor. So, if both are then zero then the numbers are equal. So that means that, well, if it's three equal to two, well, that it would be the case if two is equal to one and that would be the case if one is equal to zero but one is not zero because one is the successor of zero so that is false so two is not equal to one so three is not equal to two. The last axiom is in a sense the most important one. It's a bit more complicated than the others. If a set, X, contains zero and for every natural number and it also contains it's successor and prime then X contains all of N or put differently N as a subset of X. This is the famous principle of mathematical induction. Okay, this is very impressive. So the numbers are one is just the successor of zero, two is the successor of the successor of zero, three is the successor successor successor of zero et cetera. So we know what the numbers are. But, how do we add and how do we multiply? For this of course we need definitions. Now the definition of addition are not that that strange. First, you define what it means to add zero to a number and of course we want that to be the number itself. Then, you give the definition of adding the successor of a number to a number and you say, "well, that is the same thing as adding the number and then taking the successor." Now, if you've ever programmed, you recognize these as recursive definitions. So let's have an example, let's try to compute the sum of three and two with these rules. Well, of course since two is not zero we have to use the second rule and say, "well this is the same thing as adding three plus the successor of one, because two is the successor of one, but by the second axiom, the second axiom of the definition, this is the same thing as adding three and one and taking the successor of that." Now we repeat the whole thing and saying, "well three plus one is the same thing as three plus the successor of zero," and now we have to take the successor of that and that is the same thing as three plus zero taking the successor of that and taking it once again the successor. Now we can use the first part of the definition and say, "well three plus zero,that is three" and then we still have to take two successors so we end up by the second successor of three which is five, and that is indeed exactly what we want it to be. Now, multiplication is more or less defined in a similar way. First you say, "well multiplying with zero, that should always give zero," and multiplying with the successor of a number, that should give multiplying two numbers and then adding N. Excuse me. So let's have an example of that. Let's take again the numbers three and two. Well, this is three times the successor of one, which is three times one plus three, by the second axiom, which is three times the successor of zero plus three, which is three times zero plus three and then again plus three. Now we have the first axiom three times zero is zero. So we've got zero plus three plus three and we assume that we already know everything about addition which adds up to six. So in this way, addition and multiplication are defined. Of course there's still a job to show that they satisfy all the rules we have about addition so that N plus N is the same thing as M plus N so three plus two is the same thing as two plus three and three times two is the same thing as two times three. There is some work we're not going to do here it has been done, it has been checked, everything works great. So this is the axiomatic definition of the natural numbers. It's nice to know about it. You won't need it in practice because you know, you learned all about these numbers in elementary school.