3/42 Getting into the Mathematics: Secret History

Ladies and gentlemen, I know that in parts one and two, I hit you over the head with some pretty intense stuff. All of the long quotes from scholarly journals, and the complex images flying by loaded with info, I expect that this video series is watched over and over, and you will keep finding new hidden things. I don't expect much of this to make sense on the first view, this video series is intended to be watched as a giant loop. So when you finish at the end, you go back to the beginning, and I think things will make a lot more sense on the second viewing. So to back up a little bit, in part two we covered the Greeks and how they called the indefinite and limitless, the Epiron, and the limited as the class of Paros. Plato, in the Philibus, talked about the Mekton, which was a mixture of the Epiron and the Paros. Kantor, following Plato's lead, built a precise mathematical scaffolding into the infinite with his set theory, which combined the limited and the unlimited into a clever mixture. I now want to quote highly respected educators Robert and Ellen Kaplan on pages 75 through 76 of their book, The Art of the Infinite, The Pleasures of Mathematics. Quote, an Agamemnon who lived 150 years before Socrates recognized the foolishness of claiming that one element or another, earth, air, fire, water, was the source of everything else. Rather, he said, the source is the Epiron, as if distinction rose out of indistinction, the way it does in so many creation myths. We think this way still, seeing speciation on a grand scale and evolving from the unspecified, in minutely differentiated tissues from stem cells. The infinite disguised as the indefinite is our only begetter, but in the same guise, it is how we imagine the world truly to be. Our hope is to find structure to the infinite, behind what may be only superficial indefinition, regularities governing infinite ensembles, powers, dominions, and thrones among its blurred degrees. Mathematics is the art of the infinite, because whatever it focuses on with its infinite means discloses limitless depth, structure, and extent. Unquote. Let us now delve into some of Cantor's actual mathematical ideas. Cantor did not think of sets as mere static collections. To him, they were like nebulous clouds, potentially engulfing countless unities. And while you could make a bijection from one set to another, you still cannot, in most cases, articulate exactly the contents of the set in some affable way. Cantor did not simply think of sets as crude collections of objects. The fact that sets arise naturally in our thoughts means that they give us an insight into the platonic world of pure, bodiless ideas, sometimes mockingly called the platonic heaven. To Cantor, sets gave crucial insight into the world of our ideas and thoughts, and the first thinker was, to Cantor, God. This is opposed to Newton's God as the first physical mover. Cantor, in a poetic moment quoting Dedekind when they were together in the Alps vacationing, talks about sets reflecting some platonic form. And the set that is the world of our thoughts is something he mentions such that for every thought you can have a thought about that thought. Also, you can count subsets of this world. You can count thoughts, and thus the natural counting numbers are born from this. Cantor imagined the history of ideas as a great stream of these thoughts. If a set can be corresponded to the full set of positive integers, also called the natural numbers or just the counting numbers, or to a subset of these numbers, it is said to be countable. As a schoolchild, we learn one example of the set of all natural numbers when we learn how to count. 1, 2, 3, 4, etc. There are some paradoxes about countable sets that go all the way back to Galileo and perhaps much earlier. For example, you can correspond the set of the positive integers with the set of the positive even integers. Now this seems counterintuitive, but it can actually be proven to be the case. Welcome to the paradox of the world of infinities. Cantor's breakthrough was to try and correspond the set of counting numbers, that is the natural numbers 1, 2, 3, etc., to the set of real numbers. Now remember that a real number can be an integer, like 2 or 5, or it can be a number which expresses a ratio, such as the rational number 1 third. Not only that, but the real numbers also include numbers with an infinite and non-repeating sequence of decimals when represented in decimal form, the so-called irrational numbers, such as pi. Cantor was able to prove that the set of all real numbers was not countable. Just take a second and think about that. You could never count the set of all real numbers. You could not correspond all of the real numbers to the counting numbers. If you attempted to do so, an infinity of a certain cardinality would be left over of real numbers that are not corresponded. Now one of a couple ways that he showed this, what is commonly called the diagonal argument, and there are YouTube videos which explain this very geometrical visual proof very clearly, but in a nutshell you can list out patterns of repeating numbers. If you're looking at the video form of this talk, you can see here in binary, you can list them all out so that the first line has all zeros, the second line has all ones, the third line has 0101, the second line has the reverse of that, and then you can methodically start to show all of the possible permutations. And eventually you have a graph that looks something like this. Now if you feel like we left some numbers out because of our technique for enumerating numbers, feel free to just add any numbers that you want. They can be infinitely long decimals. You can just keep adding them to the end of the list. It really doesn't matter how many numbers you add to this two-dimensional grid. Now take the diagonal of this and you have another new number. It can be easily demonstrated to our intuition that if you switch the zeros and the ones in this new diagonal number, you have a number that clearly is not part of the numbers in the graph anywhere. And this is obvious because you can go to the nth place where the diagonal lies in that particular number and you can show that in that particular one case, that number, because we switched it from a one to a zero or vice versa, is different than our original number. Thus all of the numbers that make up this diagonal have at least one number different. Thus you can show an example of one of the many numbers not in this countably infinite graph of repeating natural numbers, in this case binary numbers. Now did you get that? Just to recap, this new number that we've created, because it is a switch of the diagonal, is clearly different in at least one position, that is the first position with the first row since we switched the number. It's also different than the second row because we switched that number in the second place and the third place for the third row and so on. So clearly this is a number that is not on this graph. It is as if the irrational numbers with their freedom to be non-repeating are infinitely greater than the infinite sets of natural numbers. Now there's something interesting going on here because on the one hand you have a way to demonstrate using an infinite string of numbers a number that is outside of an countably infinite set of numbers. You could have a Turing machine which can perform infinite tasks, countably infinite tasks, that is create this number much in the same algorithm that we use to create this number. However at the same time the existence of higher cardinalities, that is infinities that is greater than the countably infinite set, shows that there are things that a Turing machine is not capable of even though it has the capability of generating countably infinite sets. And as we will see later on this begins to shed some light on the concept of non-computability. Of course to call a Turing machine a machine at all is to stretch the metaphor. It is a machine that only exists in pure concept. The infinity of real numbers is so big that the vast infinity of the natural numbers appears to dwindle into nothingness in comparison with the set of real numbers. But the infinity of the natural numbers is quite remarkable in and of itself. David Hilbert, younger than Cantor but more famous in his day, showed that if you have a hotel with a countably infinite number of rooms and they are all occupied you can always make room for more guests by having all of the guests get up and move over to the next room or however many rooms over you need to create whatever vacancy is required. If you think about this, this is pretty strange. If you already have every single natural number already matched up with every guest, they're in a one-to-one correspondence or bijection. Let's say in the form of room numbers. You can still add more guests and somehow another natural counting number is there ready to be matched to these new guests. A rather remarkable thing when you think about it. It seems as if in the universe of mathematics there is no finite supply of natural numbers so there is no need to be stingy with them. As you will see as soon as you need one, one appears for you. It is an endless cornucopia. But if the infinity of the real numbers was greater than the infinity of natural numbers, there must be a number greater than the last infinite natural number. Cantor discovered the first transfinite ordinal number for which he used the greek letter lowercase omega to represent. So omega represents the number that comes just after all of the infinite natural numbers also known as the finite ordinal numbers one two three four and so on and then you get to the end and then you have omega. Now in the hotel example you can think of the natural numbers as being all of the room numbers. So the first transfinite ordinal number omega is the number of the room that comes just after the last room you can count. Are you still with me there? All right maybe that was a confusing example because the rooms as I know them are countable. The room with the number omega is not like any other room that I can conceive. Not in this universe at least. Pretty weird. So far we have defined the transfinite ordinal number. So a good way to think of ordinal numbers is that they sound sort of like ordered numbers. That is they have a hierarchical rank. There is no ambiguity between which one comes before or after between any two distinct members of such a set. You have the countably infinite finite ordinal numbers one two three four etc and then you have beginning with omega the transfinite ordinal numbers. If we instead think of these different infinities as being of a given power or to use common language size in comparison with each other we can use a different sequence of numbers. They are the transfinite cardinal numbers. But again cardinality to Cantor goes deeper than just to think of it as its size. Cantor is careful to note that cardinality and ordinality are themselves mental abstractions from sets themselves. As Cantor writes again from We will call by the name power or cardinal number of set M the general concept by which by means of our active faculty of thought arises from the aggregate M when we make the abstraction of the nature of its various elements and of the order in which they are given. While cardinality may be determined by ordinality, ordinality is never determined by cardinality. And this is easy to see because you might have the set of two and three and you might have another set of three and two. They are different sets. Their order is different. However they both have the same cardinality. So obviously cardinality has no effect on the ordinality. Now we've been talking about The infinite set of natural numbers that we have been looking at can be ordered in more than one way. You can picture the sequence of counting numbers going forwards or even backwards and quite a few different other possible orderings. In fact the number of different ways you can arrange the counting numbers is uncountable. Wow that does more than 10 right?