 In his dialogue concerning two new sciences, Galileo Galilei made a rather heretical observation. The perfect squares are a very small portion of the whole numbers. For example, there are only 10 perfect squares between 1 and 100, but there are 100 numbers. But every number has an associated perfect square. For example, the number 1357 is associated with the number 1357 squared. So which is it? There are fewer perfect squares than whole numbers, or there are just as many perfect squares as whole numbers. Galileo's conclusion equal, greater, and less only apply to finite sets. And this radical idea got him into trouble with the authorities. Well, his support of the heliocentric model also played a role. But let's see if we can make some sense out of infinity. So, Vickard Dedekind gave the following definition, a set is infinite if there is a one-to-one correspondence with one of its proper subsets. So let's show that the set of natural numbers is infinite. So definitions are the whole of mathematics, all else is commentary. We have Dedekind's definition of infinite. And so we want to try and find a one-to-one correspondence between the natural numbers and one of its proper subsets. So what we need is a rule that matches every natural number with a unique other natural number in such a way that not all natural numbers are used. So let's think about that. We have our listing of natural numbers. And one possibility is to offset the match. So we can take the list of natural numbers and offset it by one. So one goes to two and back, two goes to three and back, three goes to four and back, and so on. And we see that every natural number is matched to the unique natural number that is one greater. And every natural number in A is matched to a unique natural number. And A is a proper subset of the natural numbers because it doesn't include one. And so the natural numbers can be put in a one-to-one correspondence with a proper subset. So the set of natural numbers is infinite. Let's add in one more idea. We define countably infinite. If a set has the same cardinality as a set of natural numbers, we say the set is countably infinite. Now suppose we can find a one-to-one correspondence between a set and the set of natural numbers. We can represent this correspondence by having our set of natural numbers and having our other set and identifying which natural number corresponds to which element of the other set. What's useful here is that we can interpret this correspondence as follows. There's a way to list the elements of S so that every element of S is somewhere on the list and every natural number corresponds to some element of S. Now in general, finding a one-to-one correspondence between an infinite set and the set of natural numbers is a matter of finding a way to list all the elements in order. The difficulty is coming up with a way to order the set. We have to be creative. And unfortunately, it is impossible to teach how to be creative. Being creative is a skill and the only way to get better at it is to practice. And most importantly, you will never be creative if you look everything up and do what other people tell you. So you should skip the rest of this lecture, but you probably won't. Let's take a different set. How about the set of integers? That's the positive and negative whole numbers along with zero. And we want to prove or maybe disprove the set of integers is countably infinite. And so the first question is, can we list the integers in order? No. There is no smallest integer. So there's no way to choose a first integer or is there? And the thing to remember is that order is where you find it. We only have to find some way to list all the integers. So we don't need to limit ourselves to a traditional numerical order. So for example, we might list the integers as 0, 1, negative 1, 2, negative 2, 3, negative 3, and so on. And this allows us to put the integers with one-to-one correspondence with the set of natural numbers. And we claim that every integer is on the list somewhere and every natural number will correspond to some number on the list. Now here's an important idea. We can make these claims and if they happen to be true, we're in good shape. But before we move on, we'll want to make sure that these are really true. Because remember, if you don't find your mistake, someone else will. So you'll want to check over your work very, very carefully to make sure you haven't made any mistakes. So we claim every integer is on the list. And so the question you should ask yourself is, can you think of an integer that isn't on the list? For example, is negative, well, this thing on the list? And if you think about how our list is organized, this negative number is going to come immediately after the corresponding positive number. So this number is right after this number. Of course, you might wonder if this number is on the list. And if you look at how our list is organized, this number is going to be the first term in the pair that's immediately after this number. Well, is this number on the list? Again, if you consider how the list is constructed, it's got to be in the pair right after this number. And so someplace way, way, way, way, way, way, way down on the list, we have this number and immediately after it is going to be this number. And so this number is definitely on our list. Now, we also may be claimed that every natural number corresponds to some number on the list. For example, the natural number 39,874. And since we have all of our natural numbers, we know that it's going to be matched up to some integer. And if you think about it, it's actually going to be a positive number and the first of a pair K, negative K. And you might get that by noticing that our even numbers are always matched up with positive numbers. And in fact, with a little effort, you can find the actual number. The 39,874th integer listed this way will be 19,937.