I have just come across the concept that there are different types of infinite numbers (whole integers, irrational numbers etc.) and the concept that for example, the amount of irrational numbers between 0 and 1 is higher than the amount of whole numbers from 1 to infinity.
I guess I just don’t understand how an infinite amount of something can be bigger/smaller than an infinite amount of something else…
Please un-f**k my brain 😀
In: 4
I think it has to do with countability. If you are able to define a set within two arbitrary points, it is considered countable. Uncountable sets are, by definition, larger than countable sets. Therefore, an infinite Uncountable set is larger than an infinite countable set.
Let’s back up a second.
A “set” is just a collection of objects. A set is finite if it has a finite number of objects. Clear, right?
What makes a finite set “bigger” than another finite set? Well, simple, you say. We can just count up the objects in each set. Whichever set has more elements is the bigger set. Another (important) way to think about this is this: Let **A** and **B** be sets. We can say that **B** is *larger* than **A** if, when we pair up each element of **A** with a unique element in **B**, there are elements in **B** that get left unpaired.
For instance, let **A = {1,2,3}** and **B = {a,b,c,d}.** We can easily see that no matter how we pair up the elements, **B** will always have an element “left over”. For instance, if our pairs are **1a, 2c, 3d**, the element **b** in **B** gets left alone. So we say **B** is larger than **A.**
Now, we want to extend this idea of “bigger” to infinite sets. Our first approach of “counting the elements” clearly doesn’t work, so we use the other approach. This leads us to define what “size” even means for infinite sets:
>**Definition: two infinite sets are the same size if we can pair up their elements in such a way that no element in either set is left unpaired.**
So we have a definition. Great. So when can we actually achieve this pairing? Well, let’s take two sets: The positive whole numbers and the positive even numbers. Intuition suggests that the set of even numbers is smaller, right?
Not so fast. Think about pairing them up as follows: **(1,2), (2,4), (3,6), (4,8), (5,10), . . .**
Under that pairing, every single even number gets paired to an appropriate natural number, so we say the sets are the same size. In fact, we can generalize this idea: If we can list out the elements of a set in a way that eventually reaches every elements, then that listing is itself a way of pairing each element of that set with a natural number. For example, we can list the rational numbers (fractions) [like this](https://demonstrations.wolfram.com/EnumeratingTheRationalNumbers/img/popup_2.png).
Clearly any set we can do this with is the same size as the set of natural numbers. We call these sets *countable.* The next natural question is this: Is every infinite set countable? Can we pair up elements in *any* set with the natural numbers just by finding clever ways of listing them?
Nope. The natural numbers, the integers, the rationals, etc. are all the same size, but it turns out the set of *real* numbers (or the set of irrational numbers) is much larger. Try as you might, there is no possible algorithm that lists out irrational numbers in a way such you will eventually get to every irrational number. You can’t do it.
This should make some kind of sense: integers, fractions, etc. can all be *represented* with a finite list of symbols, and since there are only finitely many *symbols,* you can just enumerate every possible string of symbols and eventually you’ll arrive at any chosen number. Irrational numbers cannot, in general, be represented this way. You’d need an infinite string of digits to represent just a single “irrational number”.
Infinity is a concept, not a number… and cannot be directly compared to them. There is no number “infinity”, it’s about how you got there.
Most commonly infinity comes up when you’re looking at long term trends and patterns. However, even if two different variables will get “infinitely large” given an unlimited amount of time, often one is still always bigger than the other. Hence, an “infinity” divided by an “infinity” could be anything, from 0 to 1 to 7 to another infinity.
If you write an equation like x^2 / 4x, then as x becomes infinitely large, you have an infinitely large top and bottom of the fraction. However you can also very clearly see that beyond x=4, the top is always bigger than the bottom and the difference just keeps growing. So here we have 2 different infinities which are also infinitely apart from each other. They’re both “infinity”, but one is very much bigger than the other.
In your “irrational numbers” example, the trick is to show that you can pair off each integer with a unique irrational number, and then showing that some irrational numbers will still be missed. Therefore, even though the number of integers is infinite, the number of irrational numbers is “more infinite” since we missed some.
Spoiler because this feels like a homework question.
>!For the sake of example, for each integer, I’m going to map it to the irrational number that’s written as “0.” followed by that integer, then that integer+1, then +2, and so on indefinitely. So irrational number 17 is 0.1718192021222324…. Clearly this will be unique for each possible integer, and the pattern should be easily understood for any possible integer, so I’ve fulfilled my obligations about building this pairing. Yet also clearly I’m missing a lot of irrational numbers, like any that would begin with `0.00`. Ergo, there are more irrational numbers between 0 and 1 than there are integers between 1 and infinity.!<
To begin with, infinity is not a number. It’s a concept that allows a lot of complex math to work. Similar to zero. You can’t have zero of something, and you can’t have infinity of something.
The different inifinites are for calculating different types of math. An infinite negative number has to be different than an infinite positive number, but neither of those “exist”.
Step 1: What does it mean for two finite groups of objects to be the same size i.e. to have the same number of objects in them?
Even if we don’t have the concept of counting, we can find out. We pair off one object from the first group with one object from the second, and remove them. Then repeat the process with the new, smaller, groups. When one group is empty we look at the other group. If it is also empty, the two groups were the same size. If there are still some objects left then that group was bigger.
Step 2: we extend this method to infinite sets. This is trickier because if we do it one by one, we never end. But we do something similar. Here is an example. Consider the set of all positive integers: 1,2,3,4… etc, and the set of all positive even integers 2,4,6,8…. Etc.
It might look like the first set has more objects in it, because it have all the same objects as the second set, plus all the odd integers. But we can map each object in the first set to exactly one object in the second set so that ever object in each set appears in one of the mappings:
1 <->2
2<->4
3<->6
And in general
n<->2n
If we can do this in some way with two infinite sets, then we say that they are the same size.
Step 3: Suppose we have two infinite sets and we can *prove* that we can’t make such a mapping between them. I.e. however we try there will always be some members of one of the sets unmapped. In that case we would say that the set with unmapped members is bigger. I.e. it’s size is a bigger infinity than the other’s size.
Here’s an outline of a proof that that set of real numbers between 0 and 1 is a bigger infinity than the positive integers:
Any real number between 0 and 1 can be expressed as an infinite decimal 0.abcdef… etc
Now Imagine we had a mapping between the two sets it will look something like this
1 <-> 0.3742851….
2 <-> 0.4188803….
3 <-> 0.9013246….
Etc
Now we can construct a real number that can not be in the list.
We choose the first decimal place to be different to the first decimal place in the first number. So here the first decimal place is 3, so we pick, say, 7.
We choose the second decimal place to be different to the second decimal place in the second number. Here that is 1, so we can pick any other digit. Eg 9
We choose the third decimal place to be different to the the third decimal place in the third number. Here that is again 1, so we can pick eg 5
And we continue for ever.
The resulting decimal is definitely between 0 and 1, but is not in the mapping because for every number n it differs from the nth number in the list in the nth decimal place.
This process would work no matter how we tried to list the real numbers, ie. No mapping is possible between them and the integers, and so the set of them must be larger.
Let’s back up a second.
A “set” is just a collection of objects. A set is finite if it has a finite number of objects. Clear, right?
What makes a finite set “bigger” than another finite set? Well, simple, you say. We can just count up the objects in each set. Whichever set has more elements is the bigger set. Another (important) way to think about this is this: Let **A** and **B** be sets. We can say that **B** is *larger* than **A** if, when we pair up each element of **A** with a unique element in **B**, there are elements in **B** that get left unpaired.
For instance, let **A = {1,2,3}** and **B = {a,b,c,d}.** We can easily see that no matter how we pair up the elements, **B** will always have an element “left over”. For instance, if our pairs are **1a, 2c, 3d**, the element **b** in **B** gets left alone. So we say **B** is larger than **A.**
Now, we want to extend this idea of “bigger” to infinite sets. Our first approach of “counting the elements” clearly doesn’t work, so we use the other approach. This leads us to define what “size” even means for infinite sets:
>**Definition: two infinite sets are the same size if we can pair up their elements in such a way that no element in either set is left unpaired.**
So we have a definition. Great. So when can we actually achieve this pairing? Well, let’s take two sets: The positive whole numbers and the positive even numbers. Intuition suggests that the set of even numbers is smaller, right?
Not so fast. Think about pairing them up as follows: **(1,2), (2,4), (3,6), (4,8), (5,10), . . .**
Under that pairing, every single even number gets paired to an appropriate natural number, so we say the sets are the same size. In fact, we can generalize this idea: If we can list out the elements of a set in a way that eventually reaches every elements, then that listing is itself a way of pairing each element of that set with a natural number. For example, we can list the rational numbers (fractions) [like this](https://demonstrations.wolfram.com/EnumeratingTheRationalNumbers/img/popup_2.png).
Clearly any set we can do this with is the same size as the set of natural numbers. We call these sets *countable.* The next natural question is this: Is every infinite set countable? Can we pair up elements in *any* set with the natural numbers just by finding clever ways of listing them?
Nope. The natural numbers, the integers, the rationals, etc. are all the same size, but it turns out the set of *real* numbers (or the set of irrational numbers) is much larger. Try as you might, there is no possible algorithm that lists out irrational numbers in a way such you will eventually get to every irrational number. You can’t do it.
This should make some kind of sense: integers, fractions, etc. can all be *represented* with a finite list of symbols, and since there are only finitely many *symbols,* you can just enumerate every possible string of symbols and eventually you’ll arrive at any chosen number. Irrational numbers cannot, in general, be represented this way. You’d need an infinite string of digits to represent just a single “irrational number”.
One of the best ways I had it described (year ago) was
* you have a fish tank with 5 fish in it
* Now, you have an infinite number of fish tanks, each with 5 fish in it
* Fish infinity > fish tank infinity
One of the best ways I had it described (year ago) was
* you have a fish tank with 5 fish in it
* Now, you have an infinite number of fish tanks, each with 5 fish in it
* Fish infinity > fish tank infinity
Let’s consider the series that happens when you add one to the preceding number: 1, 2, 3, 4, 5, 6, 7, etc. etc. etc.
We know this series will go on to infinity. We also know that we can calculate it’s discrete value at any point along the way.
So far so good?
Now let’s compare that to the series where we double the preceding value: 1, 2, 4, 8, 16, 32, 64, 128, etc. etc. etc.
We know that this series will *also* go on to infinity, and we can calculate a discrete value at any point along the way.
Which of these two series is “bigger”? Again, we know that both will continue to infinity, but one of these definitely approaches infinity at a *much faster rate* than the other.
Step 1: What does it mean for two finite groups of objects to be the same size i.e. to have the same number of objects in them?
Even if we don’t have the concept of counting, we can find out. We pair off one object from the first group with one object from the second, and remove them. Then repeat the process with the new, smaller, groups. When one group is empty we look at the other group. If it is also empty, the two groups were the same size. If there are still some objects left then that group was bigger.
Step 2: we extend this method to infinite sets. This is trickier because if we do it one by one, we never end. But we do something similar. Here is an example. Consider the set of all positive integers: 1,2,3,4… etc, and the set of all positive even integers 2,4,6,8…. Etc.
It might look like the first set has more objects in it, because it have all the same objects as the second set, plus all the odd integers. But we can map each object in the first set to exactly one object in the second set so that ever object in each set appears in one of the mappings:
1 <->2
2<->4
3<->6
And in general
n<->2n
If we can do this in some way with two infinite sets, then we say that they are the same size.
Step 3: Suppose we have two infinite sets and we can *prove* that we can’t make such a mapping between them. I.e. however we try there will always be some members of one of the sets unmapped. In that case we would say that the set with unmapped members is bigger. I.e. it’s size is a bigger infinity than the other’s size.
Here’s an outline of a proof that that set of real numbers between 0 and 1 is a bigger infinity than the positive integers:
Any real number between 0 and 1 can be expressed as an infinite decimal 0.abcdef… etc
Now Imagine we had a mapping between the two sets it will look something like this
1 <-> 0.3742851….
2 <-> 0.4188803….
3 <-> 0.9013246….
Etc
Now we can construct a real number that can not be in the list.
We choose the first decimal place to be different to the first decimal place in the first number. So here the first decimal place is 3, so we pick, say, 7.
We choose the second decimal place to be different to the second decimal place in the second number. Here that is 1, so we can pick any other digit. Eg 9
We choose the third decimal place to be different to the the third decimal place in the third number. Here that is again 1, so we can pick eg 5
And we continue for ever.
The resulting decimal is definitely between 0 and 1, but is not in the mapping because for every number n it differs from the nth number in the list in the nth decimal place.
This process would work no matter how we tried to list the real numbers, ie. No mapping is possible between them and the integers, and so the set of them must be larger.
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