I'm trying to think of a way to intuitively explain myself what a countable set means.
So far as I understand it, in very simple words, a set $S$ is countable iff you can "name" or "list" down each and every member of $S$, so in a way, you've "counted" the elements of $S$ by "naming" them.
This intuition is directly inspired from the definition that $S$ is countable iff $S = \varnothing$ or $\exists$ a surjection $f : \mathbb{N} \to S$ so $f$ is a sequence in $S$ and thus you list down the members of $S$ as the sequence $\left(f(n) \right) = \left( f(1),f(2), \ldots, f(n), \ldots \right)$
Before, I go ahead. Let me know if this "way" of thinking about countable sets is correct or does it have any flaws?
Following that intuition, would it be correct to say that $S$ is countable if $S$ is a set of discrete points. What I mean by discrete is that there exists a neighborhood $N(s)$ around each members $s$ of $S$ such that $N(s) \cap \{s \} = \{s\}$.
Now, I know that "discreteness" is a topological property and it has nothing to with countablity, but hey, I'm trying to look for an intuition so think of $S$ (in this case) as some kind of metric space or something.
The only flaw, I see with the latter intuition is that if $(S,d)$ is a discrete metric space, i.e. $d(x,y)=1$ if $ x=y$ and $0$ otherwise, then any set $S$ now becomes countable, for e.g. if $S= \mathbb{R}$ but cardinality of $\mathbb{R}$ is not the same as that of $\mathbb{N}$, so this intuition changes the cardinality itself. I mean cardinality is cardinality, the number of elements in a set shouldn't change no matter what, right? I feel like discreteness $\Rightarrow$ countability but converse doesn't hold maybe or something. So this intuition is lacking something, it would be very helpful if someone could complete or fix this intuition for me. Thanks!
Any other kinds of intuitions are also very welcomed.
Yes, this is correct. It's right there in the name: a set is countable if you can count it.
This is not quite right. The problem is that any set, arbitrarily large, can be equipped with the discrete metric, or the discrete topology.
However, what is true is that if a subspace of $\mathbb{R}^n$ equipped with the subspace topology is discrete, then it is at most countable, and this is a nice exercise. So, loosely speaking, uncountable discrete spaces are "too big to fit into Euclidean space."
I wrote this in a comment but it's worth writing in the answer: a pretty good intuition is that a set is countable if its elements can be described using "a finite amount of data." In fact you can prove that the set of finite words on any finite alphabet (an "alphabet" is just a set and a "word" is a finite list of elements of that set) is countable, which is a very nice exercise and a pretty good way of making rigorous sense of the meaning of "a finite amount of data." For example the set of English sentences, or Python programs, or $\LaTeX$ formulas, is countable.
On the other hand, speaking somewhat informally, the real numbers are uncountable because to specify an arbitrary real number you need to specify infinitely many digits, which is "an infinite amount of data." Checks out!
The uncountability of $\mathbb{R}$ has the following somewhat disturbing implication: only countably many real numbers are describable by English sentences, Python programs, or $\LaTeX$ formulas. You can check out the Wikipedia articles on computable numbers and definable numbers for more on this.