Cardinality in mathematics
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Waaaay back when numbers were first invented, we only had natural numbers. Not even 0 was included. The numbers represented a count of things. I have 5 apples. There are 16 people here. Things here is used as an abstract term. Even if I eat half an apple (or person), I don’t have half of a thing, I just have 4 apples and 1 half apple, I still have 5 things.
Cardinality is the measure of the number of elements of the set. Here we have to alter my earlier, simplistic definition of a thing. Modern mathematics is built on Zermelo-Fraenkel Set Theory. There, it is not allowed to repeat items within a set. If there are 2 identical items in a set, they are only considered to be one item. So cardinality is more like the number of unique elements in a set. In terms of apples and people, no two are alike, so as long as you can uniquely identify each element of the set, you can count their number. The value you end up with is the cardinality. For people, you can use names. Instead of 3 people, the set contains Andrew, Brenda, and Cherie.
We also introduce 0 here because a set can have no elements. This is how we get the whole numbers. Still no negative numbers or fractions because a set can’t have negative of parts of an element, it can only contain “whole” elements.
So the next part is not necessarily important to cardinality itself, but just an insight about the rigor that went into modern mathematics. Built on Zermelo-Fraenkel Set Theory, we have a list of axioms that are fundamental to modern math. One of the axioms basically says “0 exists.” We call it the null set or empty set, an its written like this: { }. Now, using the axiom of pairing and the axiom of union… probably one other, we can take a thing that exists and put it in a set. The set containing the empty set (written as { { } }) is a set with a cardinality of 1, because it contains 1 element.
And in fact, this is the strict definition of the number 1 in modern mathematics. It’s just that the average person never needs to know that there is a concrete, axiom-based definition of 1 because it’s so obvious that it doesn’t need one. Unfortunately, mathematics doesn’t accept “it’s just so obvious” as a formal proof. The successor to one is two. Can we make a set with two elements. And remember, they must be unique elements. Well, 1≠0 and we have proven that both 0 and 1 exist. So 2 looks like the set containing both the empty set and the set containing the empty set: { { } , {{ }} }. This set has 2 elements so its cardinality is 2.
Just one more and I’ll leave it. 3 is the set containing all three of the empty set as well as the set containing the empty set, and lastly, the set containing both the empty set and the set containing the empty set: { { } , {{ }} , { { } , {{ }} } }.
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