How are some infinities bigger than others?

725 views

I was watching Veritasium’s video about math having a fatal flaw. He explained that if we make a set of all the numbers in between 0 and 1, then added one to the first digit of the first number and added one to the second digit of the second number etc, we would always have a new number. He said this proved that there were more numbers in between 0 and 1 than natural numbers.

I was confused as to why you can’t do this with natural numbers, and how that proved one infinity was smaller than another.

In: Mathematics

16 Answers

Anonymous 0 Comments

[deleted]

Anonymous 0 Comments

>I was confused as to why you can’t do this with natural numbers,

If you take a natural number, and change one of its digits after the decimal place, you don’t get another natural number. If you start with, say, 3, and change the ten thousandths place, you get 3.0001, which isn’t a natural number— so it’s not a problem that it’s not found on your list of all the natural numbers.

Anonymous 0 Comments

[removed]

Anonymous 0 Comments

> He explained that if we make a set of all the numbers in between 0 and 1

Not just a set, a list. It has to be well-ordered for the proof to work. There has to be a first number and a second number and so on. They can’t all be in a disorganized heap.

Remember, the point of the proof is to show that such a list is *impossible.* The argument is, no matter what list you present, his method will find a Real number 0 < x < 1 that isn’t on the list.

Anonymous 0 Comments

The thing about math is, in some ways, it’s like a game. The rules are whatever we say they are. Math is a human invention so all of its rules and definitions are whatever humans have come up with.

So we have decided that, in math, we can have things called sets. Sets are just like collections of things. And sets can have sizes, which we define by the number of things in that set.

But what about the size of sets that have an endless amount of things? Like the set of all natural numbers? Or the set of all real numbers?

Well, in this case we have decided that to sets with an endless amount of things are the same size if you can create a rule that maps every single object in each of the sets on a 1-to-1 basis without missing any in either set.

For example, take the natural numbers {0, 1, 2, 3….}

Then take the even numbers {0, 2, 4, 6…}

Clearly there are “more” natural numbers than even numbers, right? Because the natural numbers *include* the even numbers, plus more!

Wrong.

Given our rule, they are the same size because we can map them 1-to-1. Take any natural number, multiply it by 2 and you get a unique even number. Taken any even number, divide it by 2 and you get a unique natural number. 1-to-1 mapping. They are the **same size.**

So we began to wonder if all sets with an endless amount of objects were the same size.

Back in the day, a dude named Cantor came up with a rather elegant argument that showed that the set of real numbers is actually bigger than the set of natural numbers. He created a proof that showed that, no matter what rule you created to map the natural numbers to the real numbers, that there would exist real numbers not accounted for in that mapping. That there would always be and endless amount of real numbers left over.

For that reason, we consider the set of real numbers as being “larger” than the set of natural numbers.

Notice that, at no point have I used the word infinity. Unfortunately, “endless” *means* “infinite” and both of those sets we were talking about (natural and real numbers) are endless, so they are both infinite, yet according to our rules of math, the latter is larger than the former. Hence, “some infinities are bigger than others.”

Anonymous 0 Comments

First, we need to agree on what a list is. A list is just a way to assign a real number to each integer. There are a lot of possible lists, so just to make things clear, here is a possible list with an obvious rule of construction :

* 1) 0.1
* 2) 0.11
* 3) 0.111
* 4) 0.1111
* 5) 0.11111
* …
And so on

This is a particular list of real numbers between 0 and 1. It contains infinitely many items (one for each integer) but it certainly does not contain all the real numbers between 0 and 1. For exemple, it does not contain 0.2

Given any list of real numbers, there is a procedure (known as diagonalisation) that will return a real number that is not on that specific list. To illustrate the procedure, let’s just take an example with a beginning of a list

* 1) 0.**1**45869735…
* 2) 0.2**3**3813848…
* 3) 0.12**1**565351…
* 4) 0.835**8**35833…
* 5) 0.1535**7**3831…
* …

Take the digits in bold (**1** **3** **1** **8** **7** …) , add 1 to each digit (2 4 2 9 8 …) , and create a new number by concatenating these new digits after a “0.” . This gives will give a number starting with : 0.24298…
(Note that this procedure always gives a number which is a real number between 0 and 1)
This new number cannot be anywhere on the list. Indeed, for any integer n, the n-th digit of that new number is different from the n-th digit of the n-th number on that list.

Now, one can answer the following question : “Can we make a list of real numbers that contains all real numbers between 0 and 1 ?” The answer is NO. Because whatever the list is, there is always at least one real number that is not on that list, which is the one created by the procedure.

This happens because there are far too many real numbers. Which means that even if you try to make an infinite list of real numbers, you will always miss some of them. So we translate that by saying that the infinity of real numbers is larger than the infinity of integers.

You can not do the same procedure with a list of integers, because the object created by the procedure of concatenating digits is not an integer.

Anonymous 0 Comments

>… Why you can’t do this with natural numbers

Well, in order to do the trick, you need an ordered list first. You need something that clearly comes after another thing, so that you can assing an ‘ID’ to a natural number the same way Vertitasium uses a natural number as an ID for the real ones.

But the thing is, what else is there but natural numbers? Nothing else is ‘naturally ordered’. You could use the alphabet, but there is no reason why it should be ordered the way it is in the first place. A being first is arbitrary.

But 1 coming before 2 isn’t, and it’s the only thing we can all agree on. It’s also the reason why it’s called a countable infinity.

Anonymous 0 Comments

>I was confused as to why you can’t do this with natural numbers

Because then they’re not natural numbers. Natural numbers *must* be integers. Let’s try “adding one to the first digit of the first number and one to the second digit of the second number etc”:

* 1 is a natural number.
* 2.1 is not
* 3.01 is not

The Natural Numbers are 0, 1, 2, 3… But there are infinite numbers between 0 and 1. And another infinite numbers between 1 and 2, and another between 2 and 3…

The smaller infinities are called “**countable**” and the bigger ones “**uncountable**”. Here’s why:

1. Count the natural numbers: OK. 0,1,2,3,4…. It would take forever to count them all. But *if you had forever, you could do it*!
2. Now, count the numbers between 0 and 1, decimal numbers allowed: Alright. 0 ….um, *what’s the next number*? Is it 0.1? No, can’t be, then you’d have missed 0.01. Is it 0.01? No then you’d have missed 0.001…

**If you try and count the numbers between 0 and 1, you can’t even get to the first number past 0 in the list**, because no matter what number you say next, you’ll have missed an *infinite* number of numbers before that one! When trying to list the numbers between 0 and 1, there’s infinities *inside* infinities. An infinite number of numbers between each of an infinite number of numbers. It’s **uncountable**.

Listing the natural numbers doesn’t have this paradoxical-explosion-of-infinities problem. There’s just a “regular” list of numbers. It’s infinitely long, but each step is well defined and not growing within itself.

Anonymous 0 Comments

reading your post I may be mistaken but I think you may have misunderstood something.

we make a set of all the numbers in between 0 and 1, then added one to
the first digit of the first number and added one to the second digit of
the second number etc, we would always have a new number. ”

what makes me think you may have misunderstood is where you say “new number”

if my understanding is correct the number itself is not “new” it may be clearer to say it is “unpaired”

the proof is a proof by contradiction and “proves” that one “infinity” is larger than others by

1. first assuming you can pair up (a bijection) the set of natural numbers and real numbers the set of both is infinite and it is assumed that it could be done.

2. it is proven that our assumption leads to a contradiction this is done by “finding” (proving there exists) a “unpaired” number after all the numbers were paired up (this was assumed)

3. the proof that the there is a unpaired number works works by applying a process to the infinite list where a number is shown to be different from every number on the list. this is done by adding some value to the infinite representation of a real number for every real number Importantly the number “created” exists and is a real number it would exist in a set of real numbers but it can’t exist on the set of paired numbers between natural and real numbers.

4. because we assumed we could pair up the all numbers of each and proved there was a unpaired number this implies that our assumption was wrong.

further because there is a unpaired real number and presumably all the natural numbers were paired then their are “more” real numbers ie its a larger set but both sets were infinite.

Anonymous 0 Comments

>I was confused as to why you can’t do this with natural numbers, and how that proved one infinity was smaller than another.

Every natural number has a finite number of digits. Every irrational number has an infinite number of digits. If you tried this with natural numbers you’d quickly run out of digits to change.