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
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”.
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