tiny thought: squares and sets of sets.

This tiny thought was inspired by exercise 1.2.4 of Stephen Abbot’s book “Understanding Analysis”, paired with the solution set given by Ulisse Mini at https://uli.rocks/abbott/.

This exercise asks: construct sets $A_1, A_2, A_3, ...$ such that $\bigcap_{i=1}^{\infty} A_i = \emptyset$ and $\bigcup_{i=1}^{\infty} A_i = \mathbb{N}$.

To paraphrase the math notation, we aim to construct sets whose intersection is the empty set and whose intersection is the natural numbers. At first, I was baffled and surprised by the simplicity of the exercise prompt, but struggled to propose such a collection of sets.

After spending some time trying to create the sets by using prime factorization and the fundamental theorem of algebra, I peeked at the solution and was humbled by its simplicity. Ulisse beautifully unrolls $\mathbb{N}$ into a square:

And thus, we can denote each row as $A_i$. Nice! I believe this is linked to the diagonalization proof strategy we will see further down the line in Abbot’s book.

I find this rather beautiful. Imagine doing this in more dimensions!

dgv