Modern mathematics is almost universally* based on set theory, so I decided to read Pinter's *Set Theory*.

Naive set theory says that a set is any collection of objects. For example, the set of natural numbers, ℕ, is the set {1, 2, 3, …}. Naive set theory runs into several problems, so most mathematicians use Zermelo-Fraenkel (ZF) set theory, so this post also uses it. The difference is not important for this post.

The intersection *A* ∩ *B* of 2 sets, *A* and *B*, is the set of objects which are in both *A* and *B*. For example,

{1, 2, 3}∩{2, 3, 4}={2, 3}

$$\bigcap_{i \in I}^{} A_i = A_1 \cap A_2 \cap \ldots \cap A_n$$

Here is a proof of an exercise in *Set Theory*.

Let {*A*_{i}}_{i ∈ I} be an indexed family of sets. Prove that

$$\bigcap_{i \in I}^{} A_i$$

is a set.

By definition, the intersection of 2 sets is a set. The associative property holds for the intersection, so

$$\bigcap_{i \in I}^{} A_i = A_1 \cap (A_2 \cap \ldots \cap A_n)$$

We already know that *A*_{1} is a set, so $\bigcap_{i \in I}^{} A_i$ is a set if (*A*_{2} ∩ … ∩ *A*_{n}) is a set. We can repeat this step, writing that *A*_{2} ∩ … ∩ *A*_{n} = *A*_{2} ∩ (*A*_{3} ∩ …*A*_{n})

Again, we see that *A*_{2} is a set, so we now need to prove *A*_{3} ∩ …*A*_{n} is a set. We repeat this step until we get *A*_{n − 1} ∩ (*A*_{n})

which is a set, so now we go backwards, seeing that *A*_{n − 2} ∩ (*A*_{n − 1} ∩ *A*_{n})

is a set, and continuing backwards until we prove that *A*_{1} ∩ (*A*_{2} ∩ … ∩ *A*_{n})

is a set, which equals

$$\bigcap_{i \in I}^{} A_i$$

Thus, $\bigcap_{i \in I}^{} A_i$ is a set. QED

```
<li id="lambda-category">Aside from set theory, there are several other
useful foundations for mathematics, such as lambda calculus and category
theory. Lambda calculus is very popular in computer science in particular,
where it is the inspiration for functional programming using languages such
as Haskell. Category theory is also really interesting because it's just
so abstract that even mathematicians (jokingly) call it <a
href="https://en.wikipedia.org/wiki/Abstract_nonsense">abstract
nonsense</a>.</li>
```