Everything Everywhere Daily: History, Science, Geography & More - Infinity and Beyond

Episode Date: July 15, 2021

The biggest thing there is and the biggest thing there can be is infinity. It literally has no bounds. In fact, thinking about infinity can quickly give you a nosebleed because our finite minds can’t really grasp just how enormous it really is. However, what if I were to tell you that there is something even bigger than infinity? Or, to be more accurate, there are infinities that are bigger than other infinities?  Learn more about your ad choices. Visit megaphone.fm/adchoices

The biggest thing that there is and the biggest thing that there can be is infinity, right? It literally has no bounds. In fact, thinking about infinity can quickly give you a nosebleed or a headache because our finite minds can't really grasp just how enormous it really is. However, what if I was to tell you that there's something even bigger than infinity? Or to be more accurate, there are infinities that are bigger than other infinities. Follow me down the rabbit hole as I investigate infinity on this episode of Everything Everywhere Daily. What if your perceptions about the past were wrong?

ThruLine is a podcast that takes you back in time to uncover the parts of the story that may have gone unnoticed. It effectively turned day into night and how it shaped the world now. Time travel with us every week on the ThruLine podcast from NPR. This episode is sponsored by audible.com. My audiobook recommendation today is The Mystery of the Aleph. Mathematics, the Kabbalah, and The Search for Infinity by Amir D. Askell. Towards the end of the 19th century, one of the most brilliant mathematicians in history, Gaior Cantor, languished in an asylum.

His greatest accomplishment, the result of a series of extraordinary leaps of insight, was his pioneering understanding of the nature of infinity. How he came to his theories and the reverberations of his pioneering work will shape our world for the foreseeable future. Cantor's theory of the infinite is famous for its many seeming contradictions. For example, we can prove that in all time there are as many years as days and that there are as many points on a one inch line as on a one mile line. You can get a free one-month trial to Audible and two free audiobooks by going to audibletrial.com slash everything everywhere or by clicking on the link in the show notes. This is an episode I was very unsure about doing.

I came up with the idea over a year ago when I first launched the podcast. The reason I was hesitant is that the ideas I'm going to talk about are much easier to accept. explain if you can illustrate them. It's just hard to do in an audio format. But I realize this is a daily show, and if I fall totally flat on my face, I can just get back on the horse tomorrow with a brand new episode. The concepts I'm going to be talking about are actually not difficult to understand. They usually are only presented in a university-level mathematics course, but there are no equations involved. It's nothing more than ideas and concepts. If you don't get some of the ideas, feel free to go back and re-listen to a section or find some other sources online which can help

illustrate the concept. My end goal is to prove to you that there are some infinities larger than other infinities. This will start in the 19th century with a German mathematician named Geyr Cantor. Throughout history, mathematicians have had a difficult time dealing with the concept of infinity. Cantnor took steps to try to rethink the very foundation of mathematics. To do this, he created what is called set theory. A set is an incredibly simple concept. A set is a collection of distinct things. A set can be made up of anything, tangible or intangible. The fingers on my right hand can make up a set. All the presidents of the United States can make up a set. Numbers can make up a set. Don't overthink the concept of a set. It really is as simple as what I just said. The things in the set

don't even have to be the same sort of things. In fact, it's so simple, you might be wondering what the big deal is. Well, as it turns out, this idea of a set is a very handy way to think about mathematics. For example, what does it mean if two things are equal to each other? Again, a very simple idea, but one which can be really tricky to answer when you get into the weeds.

In set theory, two sets are equal to each other if there is a one-to-one correspondence to the members of each set. There is a one-to-one correspondence between the number of stooges and the number of wise men. Likewise, there is a one-to-one correspondence between the number of deadly sins, the number of Akira-Kurosawa samurai, and the number of Snow White's dwarfs. It also helps us to find what a number is.

Five is equal to any set, which has the same number of objects as I have fingers on my right hand. The number five represents the cardinality of a set with five members. So far, this is all really simple, so simple that you might wonder what the point is. It actually seems rather obvious. Well, a set can also have an infinite number of members, and this is where things get interesting. We can also use the same techniques to determine if things are equal on sets of infinite size. The infinite set that we'll look at first is the simplest such set, the set of natural numbers. This is just the set of the normal counting numbers that you're used to, one, two, three, four, five, six, etc. If you start playing with this

set, then interesting things happen. For example, if you add any finite number to that set, the size of the set stays the same. To illustrate this point, let's assume that there's a hotel with an infinite number of rooms. You go to the front desk and ask for a room, but the front desk manager says that they're sold out. There are an infinite number of people filling up the infinite number of rooms, and they are sorry, but there's no rooms available for you. No problem, you say. Just have everyone move up one room. The person in room one goes to two, the person in two goes to three, etc. Now, room one is freed up and there's a place for you to stay. You added a person, and still the infinite number of rooms have an infinite number of people. Nothing has changed.

Now let's assume that an infinite number of people show up asking for a room. Again, all the rooms are sold out. This time, you can find space for an infinite number of people by having everyone move to the room, which is twice the number of the room they were previously in. Room 1 moves to Room 2. 2 goes to 4, 3 goes to 6, etc. Now everyone simply goes to the odd-numbered rooms, and you have space for an infinite number of people. Likewise, you can subtract a set of infinite size, and it won't change the original infinite set.

For example, all of the odd numbers equally match up to the set of natural numbers 1 to 1, yet the odd numbers are half of the numbers. Any set which equals the set of natural numbers is called countably infinite. Now, before I said that a number reflects the size of a set. Five represents the size of a set with five things. Well, what number represents this infinite set? You may have heard that infinity isn't a number, and that's true.

Kennor created a new number called alif null, or alf zero, to represent the number of this infinite set. Aleph is the first letter in the Hebrew alphabet, and he picked it because they were running out of good Greek letters to use in mathematics. This is called a trans finite number, and as you can tell from my above examples, it doesn't behave like a regular number. You can't just stick it into an equation like you would three or seven. Kenner then wondered if the set of natural numbers was the same as the set of all fractions, known as the rational numbers. There is an infinite number of fractions between any two points on a number line because you can always just divide by two. Through a very clever technique, can nor figured out that the set of all fractions was equal to the set of natural numbers. It was countably infinite.

Imagine a graph with numbers going up the vertical axis and numbers across the horizontal axis. You can draw a line that connects every fraction. You snake your way connecting the dots starting at one over one, then you go to. to 2 over 1 and 1 over 2, where the numerator and the denominator add up to 3. Then you go to 1 over 3, 2 over 2, and 3 over 1, where the numerator and the denominator add up to 4. You can continue this indefinitely. All of the fractions are on a line, and hence are countably infinite. This was a surprising result, but the next result was even more surprising. The next question Kentnor had was if the set of natural numbers was equal to the set of real numbers. Real numbers are all of the

numbers which can and can't be expressed as a fraction. A real number is expressed as an infinite string of digits. Pi is a real number, as is the square root of two, and any infinite random string of digits to the right of the decimal point. The proof that Cantner came up with rocked the world of mathematics. It was probably even more clever than the technique he came up with for the rational numbers. And this is the part that might be hard to visualize, so listen closely. You can't really put infinitely long strings of numbers into any sort of order. So can't or just assume that you could make a list that was complete?

It didn't matter what the order was, but you could make an infinitely long list that had every real number on it somewhere. Just so you're visualizing it correctly, it's an infinitely long list, and every entry on this list is an infinitely long string of numbers. With every possible number accounted for on the list, supposedly, Cantnor then set out to create a new number. On this list, he took the first digit after the decimal point in the first number and added a 1 to it. If the number was a 1, it becomes a 2. If it's a 2, it becomes a 3, and if it's a 9, it becomes a 0.

He then took the 2 digit in the 2 number and added a 1 to it. He then took the 3rd digit in the 3rd number and added a 1 to it. you can go diagonally down this list changing the nth digit to the nth number. What you wind up with is a brand new number that cannot be on the list. Any number on that infinitely long list has to differ from this new number in at least one digit. The problem was the stipulation was that the list was complete. It was supposed to have every number on it. And this is called a proof by contradiction.

The implication of this contradiction was that the set of real numbers is bigger than the set of natural numbers. It is not countably infinite. This means that there is an infinity larger than infinity. The set of real numbers was given the transfinite number of LF1. As I mentioned, this realization rocked the world of mathematics. Many people could not get their heads around it, even though it was true. This diagonal proof method was used by Kurt Godel to prove his incompleteness theorem, which rocked the world of philosophy, when he proved that in any logical system,

there are statements that cannot be proven. If you think the idea that there is an infinity larger than infinity is weird, it gets weirder. There are an infinite number of infinities, each larger than the next. One of the biggest outstanding problems in mathematics deals with these infinitely large sets. It's called the Continuum Hypothesis, and it states that there are no infinite sets between alf-0 and al-F-1 in size. Mathematicians think this is true, but so far it has never been proven. So, yeah, there are infinities larger than some other infinities. The key to understanding this is the diagonal proof.

It doesn't require any advanced mathematics, just understanding the idea. And there are many resources online to help you better understand the diagonal proof. if you want to go into it some more. If you take the time to really understand it, and it's really not that hard, then you two will be able to prove to your friends at the pub that there is something actually larger than infinity. The associate producer of Everything Everywhere Daily is Thor Thompson.

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