Are these the same concepts?
I am particularly interested in space-time with a discrete structure and the like. I heard that there is not a standard way for quantizing space-time yet. At any rate, I would like also to know in a quantized space-time, coordinates merely take discrete values? Or they take discrete values only in some special quantum mechanical conditions?
Defintion of a discrete set: click
Definition of a countable set: click
Finite sets are both, countable and discrete.
The set $$ S := \{ \frac{1}{n} ~ | ~ n \in \mathbb{N} \} $$
is countable and discrete.
The set
$$ S' = \overline{S} = S \cup \{0\} $$
is countable but not discrete.
Proof: let f: $S' \rightarrow \mathbb(N) $ with $f(0) = 1$ and $f(1/n) = n+1$ then f is injective. Thus $S'$ is countable. However, $S'$ is closed and has an accumulation point $0 \in S'$. Thus it cannot be discrete (see definition).
I would understand discrete such that there is a minimum distance between the nearest element. A cubic lattice is an example of this.
In mathematics, the expression countable means that there exists a mapping from the natural numbers to the set you describe. This means that you can sequentially number every point. In a cubic lattice, this is certainly possible with an appropriate scheme. So a cubic lattice (as in an ordinary solid crystal) would serve as a countable and discrete space.
The rational numbers $\mathbb Q$ are also countable. Yet I would not call it discrete because you can always find (countable) infinitely many numbers between any two numbers. If you would make a spacetime which was $\mathbb Q^{1,3}$ instead of the usual $\mathbb R^{1,3}$ it would be countable but I wouldn't call it discrete.
A discrete spacetime is used in K. Wilson's lattice field theory where it serves as a regulator. The coordinates there do take discrete values, you just number the lattice sites with numbers from $\mathbb N^4$.
In quantum field theory, spacetime is not quantized. It is also uncountably infinite.
About the “no standard way”: Discretizing the space with a lattice is a computation aid. It has no physical interpretation. It is just darn useful when doing simulations on a computer!
There are thoughts that spacetime might be discrete in a physical way. One of this is quantum loop gravity which discretizes space at the order of $10^{-34} \, \mathrm m$. This is so small that nobody has any idea what could happen there.
"Countable" and "discrete" are separate concepts. The Cantor set, for example, is discrete, but not countable.
As for discretizing space-time, the only theory I'm aware of that takes this approach is known as loop quantum gravity. String theory, if my passing understanding is correct, does not discretize space-time in this way.
