 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. And 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 for example, let's prove the set of natural numbers is infinite. Definitions are the whole of mathematics, all else is commentary, and so are definition. And we want to produce a one-to-one correspondence between the set of natural numbers and a proper subset. So we'll set down the natural numbers. Wait a minute. The natural numbers by the paenoaxioms include zero. So we should have written this down. And we see here that we can find a one-to-one correspondence between the numbers in the top row and the numbers in the bottom row. The complicated way we might do that is this way. And since the set of positive integers is a proper subset of the set of natural numbers, then the set of natural numbers is infinite because there's a one-to-one correspondence between the set of natural numbers and a proper subset. We'll introduce one more idea. If a set can be put in a one-to-one correspondence with a set of natural numbers, the set is countably infinite. And we'll explain the countably later. The preceding shows the set of positive integers is countably infinite. Since two sets have the same cardinality whenever there is a one-to-one correspondence between them, then any set we can put in a one-to-one correspondence with a set of positive integers is infinite. Here's another viewpoint. Suppose we can put our set in some order. So we have a first element, a second, a third, and so on. This gives us a very natural way of putting the two sets into one-to-one correspondence and this suggests the following. A set is countably infinite if we can list it. So let's show that the set of integers, positive and negative, is countably infinite. Now we need to find a way to list all the integers. And so our first thought might be, well, let's just list them, 0, 1, 2, 3, 4, and so on. But this won't work since we never get to a point where we can list the negative integers. So we have to include those negative integers. So one possibility, what if we switch back and forth? So 0, well there is no negative 0, 1, negative 1, 2, negative 2, 3, negative 3, and so on. And this would give us all of the integers as a list. And so we have our list of integers. We make the correspondence with the set of natural numbers. And here the natural number n is matched to the integer. Well, I won't do all your homework for you. You can figure out a formula for that. But the important thing is that we do have this one-to-one correspondence, so the set of integers is countably infinite. These examples should show that showing a set is countably infinite relies on a clever way to list the elements. So let's try one that's a little bit more difficult, but has some very far-reaching consequences. Let's try to find the set of all written numbers and show that it's countably infinite. Now, definitions are the whole of mathematics. All else is commentary. We should define what we mean by written number. And so we'll say that the number can be written if it can be expressed in words. So a number like 6, 42, or 11,247,801 half. So here's idea number one. Since they're words, let's alphabetize them. So there are no number words that begin with a, b, c, or d. There are, however, number words that begin with e. And the first of these words is 8. Now, 8 is the first word in our list if we list the numbers in alphabetical order. After it, well, there's a bit of a problem. So we might consider the letters that could go after the t. And we think about that a little bit. And I could put billion after it. 8 billion would come after 8. And in fact, 8 billion billion would come after that. Then 8 billion, billion, billion, and so on. And since we can keep adding billion, we'll never get past the 8s. So let's try a different idea. Since the problem is that we could just keep adding billion, we'll need to limit the length of the word. And so suppose we organize the writings by both the length first and then by the first letter. So there are no number words with one or two letters. The three letter numbers are, and if we put these in alphabetical order, one is the first. And then the others are 6, 10, 2. Let's go on to the four letter number words and put those in alphabetical order. And five letter words and so on. And we can put these in a one-to-one correspondence with the natural numbers. And since every written number is going to be someplace on this list, the longer the written word is, the further down on the list it is, but every number that we can write is there. And so we can say the number of numbers that can be written out is countably infinite. Now, the strategy of this sorting by length and then alphabetizing within the length is actually useful. Alan Turing defined the following, a computable number is one that can be computed by a finite length computer program. And let's consider all of those computer programs. We can sort the computer programs by length and then alphabetize them within the given length. And so that tells us the number of computable numbers is countably infinite. This type of logic actually applies to anything we can write. Since every proof has to be written out using a finite set of symbols, we can sort the provable statements by the length of their shortest proof. Then alphabetize within the given length. And so that tells us the number of provable statements is countably infinite.