Domain Theory and Polymorphism
There are many ways to model polymorphism via domain theory, let me just describe one that is easy to understand, so you can think about it yourself. It's a "PER model".

Take any model of the untyped $\lambda$-calculus, for instance a domain $D$ such that $D \to D$ is a retract of $D$ (for instance, take $D$ such that $D \cong \mathbb{N}_\bot \times (D \to D))$. Let $\Lambda : D \to (D \to D)$ and $\Gamma : (D \to D) \to D$ be the retraction and the section, respectively.

We are going to interpret the types as partial equivalence relations (PER) on $D$. Recall that a PER is a relation which is symmetric and transitive, but it need not be reflexive. To each type $\tau$ we thus assign a PER $\sim_\tau$. Think of $x \sim_\tau x$ as "$x$ is an element of $\tau$" and $x \sim_\tau y$ as "$x$ and $y$ are equal as far as $\tau$ is concerned".

We can have some basic types (but need not), for instance if we make sure that $\mathbb{N}_\bot$ is a subdomain of $D$ via an embedding $\iota : \mathbb{N} \to D$ then we may define $\sim_\mathtt{nat}$ by
$$x \sim_\mathtt{nat} y \iff \exists n \in \mathbb{N} . x = \iota(n) = y.$$

Given PERs $\sim_\tau$ and $\sim_\sigma$, define the function space PER $\sim_{\tau \to \sigma}$ by
$$x \sim_{\tau \to \sigma} y \iff
  \forall z, w \in D . z \sim_\tau w \Rightarrow \Lambda(x)(z) \sim_\sigma \Lambda(y)(w)$$

The terms are interpreted as untyped $\lambda$-calculus terms as one would normally interpret them in $D$.

Here's the punchline. You may interpret polymorphism as intersection of PERs, that is:
$$x \sim_{\forall X. \tau(X)} y \iff
  \text{for all PERs $\approx$, $x \sim_{\tau(\approx)} y$}.
$$
We can caclulate the PER corresponding to $\forall X. X$: it is the intersection of all PERs, but that is going to be the empty PER. Calculating $\forall X . X \to X$ is an interesting exercise. Calculating $\forall X . X \to X \to X$ is a difficult exercise (which kept me busy for a week when I was a student of domain theory).

If we want recursion in our language then we need to account for fixed points. One possibility is to restrict PERs to those which contain $\bot_D$ and are closed under suprema of increasing chains. More precisely, take only those PERs $\approx$ for which


$\bot_D \approx \bot_D$, and
if $x_0 \leq x_1 \leq x_2 \leq \cdots$ and $y_0 \leq y_1 \leq y_2 \leq \cdots$ are increasing chains in $D$ such that $x_i \approx y_i$ for all $i$, then $\sup_i x_i \approx \sup_i y_i$.


We can now interpret $\mathtt{fix}_{\tau} : (\tau \to \tau) \to \tau$ by applying the Kanster-Tarski theorem about existence of fixed points, just like we do in ordinary domain theory. This time, $\forall X . X$ is not empty, as it contains precisely $\bot_D$.