why is $L=\{\{x\mid x

Let the set $L$ be definded as$$L=\{\{x\mid x<q(i)\}\mid i\in\mathbb{N}\},$$

where $q(i)$ is some bijection from $\mathbb{N}$ to $\mathbb{Q}$.

Clearly, every member of $L$ is neither an empty set nor $\mathbb{Q}$.

Every member of $L$ is closed downward.

Every member of $L$ has no largest member.

So, all members of $L$ are Dedekind cuts.

If $L$ is the set of all possible Dedekind cuts, then real numbers would be countably infinite. So, my question is, why is $L$ not the set of every possible Dedekind cut?

Edit:

I'm aware that irrational numbers are not included in $L$.

So to clarify my question, how could $\{x\mid x^2<2\lor x<0\}$ be not in $L$ although $L$ has all possible cuts corresponding to every single rational number?

I'm looking for explanation, not a counter example.