Cardinals are the numbers that let us describe how many elements are in a set. For normal people, the only cardinals that matter are the natural numbers (i.e. 0, 1, 2, etc.) and perhaps the first infinity, but for mathematicians, there are many, many more cardinals to play with. While there are literally countless cardinal numbers, it helps to begin with some small and familiar ones before we work our way up to larger numbers.

The smallest of all cardinal numbers are the natural numbers. These are used constantly in everyday life; you have two legs, the world has 7.8 billion people in it, there are 12 eggs in a carton. We use cardinal numbers so often that we often don’t think about what they mean; we hear that there are 3 people in a room and we already have an image of what that means. However, it helps to think a little more about what it means to use a cardinal.

When someone says there are four coins in a bag, this means that if you take the coins out, counting as you do, you will count up to four. This fact seems trivially obvious — and it is — but it forms the basic definition of the cardinality of a set. Imagine an infinite line of numbers painted on the ground starting at one and going up by one at each step. If you place one element from your set on each number starting at one, the number you place the last element on will be the cardinality of your set.

This initial definition is a great start, but it can only handle “countable” sets, i.e. the sets that you could count if you had an arbitrary amount of free time on your hands and didn’t get bored. However, many sets exist that you could never count even with an infinite amount of time. For example, using a technique created by Georg Cantor in 1891, we can prove that the real numbers are not countable.

Suppose you’ve come up with what you think is a way to line up the real numbers between 0 and 1 with the naturals and prove they’re countable.

1 | 0.31415…
2 | 0.12345…
3 | 0.22222…
4 | 0.00100…

Now make a new number with a different first digit than your first number, a different second digit than your second number, and so on. For example, 0.43451…. This number won’t be anywhere in your list because it differs in at least one digit from every real number you listed. Therefore, there are more real numbers between 0 and 1 than there are natural numbers. This proves that there’s more than one kind of infinity. In fact, you can make a similar argument to prove that there are infinite varieties of infinity, each larger than the last, but we’ll skip that proof for the sake of brevity.

So our original definition of the cardinality of a set isn’t complete, but it’s almost there. To define the cardinality of any set, we need to temporarily stop trying to assign a number to be the cardinality of a set and think of what it means for two sets to have the same cardinality. Based on our original definition of cardinality for countable sets, two sets to have the same cardinality if you can line them up next to each other so that every element from the first set is next to exactly one element from the second with no leftover elements in the second set. In more mathematical terms, given two sets, A and B, |A|=|B| (|A| is the cardinality of set A) if and only if there exists a bijective mapping from A to B. Additionally, we can say |A|≤|B| if and only if |A|=|C| for some subset of C of B.

Now that we know what it means for two sets to have the same cardinality, we have everything we need to define every cardinal number. If we put all the sets with equal cardinalities into boxes, called equivalence classes, the cardinal numbers are just the names of those boxes*. Three is the name of the box of sets with three elements, |N| (the cardinality of the natural numbers) is the name of the box that contains all the sets we can line up with the cardinal numbers, and |R| is the box that contains all the sets that have the same cardinality as the real numbers.

*Technically this also isn’t the true definition for cardinal numbers — the real one involves another way of measuring the size of sets, the ordinal numbers — but it’s good enough for now.

Naming the size of every set is great, and very useful, but wouldn’t it be fun if we could do arithmetic with these new measurements just like the old-fashioned natural numbers? If you answered yes, you’re in luck (if you answered no, please go back to the engineering school). There are ways to extend addition, multiplication, and division from the natural numbers to the cardinals while preserving their original properties in the naturals. However, things get a bit weird when we start talking about numbers larger than we can count, as they usually do. Adding infinity to infinity doesn’t always equal two infinities, but we’re getting ahead of ourselves. First, we need to actually define our operations. We’ll start with multiplication since it’s more convenient.

The very first definition of multiplication you were probably taught had to do with how many squares were in a rectangle that was x square by y squares. This is actually a very accurate definition of multiplication for sets. We obviously can’t use “squares”, but we can use a similar concept. First of all, instead of the natural numbers x and y, use the sets A and B containing every natural number from 0 to x-1 and 0 to y-1 respectively, and instead of squares, use ordered pairs of the elements of X and Y, for example, (0, 0), (3, y-1), etc.. These ordered pairs are elements of what is called the Cartesian product of the sets A and B. The Cartesian product (written A×B) contains every possible ordered pair of elements from X and Y, or, thinking back to squares, the position of the top right corner of every square in our rectangle in the plane. The Cartesian product obviously contains the same number of elements as the number of squares in our rectangle, and it works for sets, so it’s a perfect way to extend multiplication to cardinal numbers. We just say that for any sets A and B, |A|⋅|B|=|A×B|.

Addition is slightly more complicated than multiplication but still relies on your intuitive understanding of addition. When people talk about addition, what they’re really talking about is combining sets of things, combining a set of three apples and a set of two apples makes a set of five apples. In other words, addition is just taking the union of sets (written as A∪B). However, there is one caveat, the sets you’re taking the union of need to be “disjoint” or else things won’t work how you want them to. {1,5} has two elements and {1,2,3} has three elements, but {1,5}∪{1,2,3}={1,2,3,5}, which has four elements because the two sets are not disjoint. So in order to extend addition to cardinal numbers, the sets we’re working with need to be disjoint. Fortunately, this is very easy to do using the Cartesian product. We can define |A|+|B|=|A×{0}∪B×{1}|, which ensures that we’re only taking the union of disjoint sets and doesn’t change the result. To demonstrate, let’s use this on our sets from before that addition didn’t work for. |{1,5}|+|{1,2,3}|=|{1,5}×{0}∪{1,2,3}×{1}|=|{(1,0),(5,0)}∪{(1,1),(2,1),(3,1)}|=|{(1,0),(1,1),(2,1),(3,1),(5,0)}|=5.

We have now defined addition and multiplication in ways that work for any set, but before we finish up, let’s play around a little. We know that our new definitions work the exact same on natural numbers as the old definitions*, but how do they work on infinities? Let’s say you want to add two cardinal numbers, κ and μ, at least one of which is infinite. If one of these two cardinals is an infinity, then |κ|+|μ| will be the larger of the two. Since addition is commutative, we can assume |κ|≥|μ| without loss of generality. Since you can alternate placing elements of κ and μ in a line next to κ to create a bijection, |κ|+|μ|=|κ|=max{|κ|,|μ|}. This seems counterintuitive, but many things are when we start working with infinities. An even weirder fact, which we aren’t going to prove here is that |κ|⋅|μ| also equals max{|κ|,|μ|}.

*We neglected to prove that our new operations maintained properties like commutativity, associativity, and multiplication by zero, but they do hold.

With these definitions and a few proofs and techniques in hand, you should go off and try to explore cardinals on your own. Cardinals are fascinating and unintuitive, but understanding them will help you later on as you learn more math. I hope this has been a helpful introduction to these intriguing numbers and I wish you the best in your future explorations of math.

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