How easy it is to see your brother's faults,
How hard it is to face your own. 
You winnow his in the wind like chaff, 
But yours you hide, 
Like a cheat covering up an unlucky throw. ~~ The Buddha

I define mathematics as the universe of statements about abstract things and combinations thereof. There are three primitive kinds of things. Countable things, cutable things and cutable things with holes in them. The last category is due the Dirk Bruere. All primitive cutable things are countable, but not all countable things are cutable.

I use square brackets for function parameters e.g., $\mathit{f}\left[x\right]$. This is because function parameter lists are syntactically distinct from algebraic groupings. There are occasions when my notation collides with the common usage, for example, commutator notation. For such situations I will use \left[\![\_,\_\right]\!].

$\left[\![ \_,\_ \right]\!]$

The most difficult chapter I have encountered in mathematics in the past few decades is Vol I, Part B, Chapter 1: Construction of the System of Real Numbers in Fundamentals of Mathematics.

A PDF save of my current notes on the chapter is: The Real Numbers: BBFSK, etc.

I adhere to the convention that the natural numbers $\mathbb{N}\equiv \mathbb{N}_1$ are self-counting. That is, when counting the natural numbers in order, we enunciate the name of the number as we count it.

$$\mathbb{N}\equiv \left\{n\in \mathbb{Z}\backepsilon n>0\right\}$$

I call the set of non-negative integers $\mathbb{N}_0$ the whole numbers. I believe that the purest axiomatic construction of the real numbers begins with the natural numbers. Even if one does not share this opinion, it is certain that an appreciation of the difference between beginning with $\mathbb{N}$ versus $\mathbb{N}_0$ is valuable.