What is the cardinality of Dana Scott's $D_{\infty}$? In the 1960's, Dana Scott constructed the domain $D_{\infty}$ which has the property
$D_{\infty} \cong D_{\infty}{}^{D_{\infty}}$.
Its construction is based on a cumulative hierarchy of infinite sequences.
For an exposition of its construction one can read the Stenlund (1972) book, “Combinators, $\lambda$-terms and proof theory", Ch1 §6.
Assume that we know the cardinality of $\lVert D_0\rVert = d$.
Then $D_1 = D_0 {}^{D_0}$, so $\lVert D_1\rVert = d^d$.
$D_2 = D_1{}^{D_1}$, so $\lVert D_2\rVert = {(d^d)}^{(d^d)} = d^{d^{(d+1)}}$.
In general, $D_{n+1} = D_n{}^{D_n}$.
Is there a way to express $\lVert D_{\infty}\rVert$ in terms of $d$?
Or, in the finite case, a nice formula for $\lVert D_n\rVert$?
 A: If you take $D_0$ to be the two-element chain, $D_1\cong(D_0\to D_0)$  is the three-element chain consisting of order-preserving endofunctions of $D_0$ (not a $4=2^2$-element set). Then $D_2$ is a lattice with ten elements  (not $27=3^3$).
It is then a combinatorial question how big the subsequent lattices are; maybe someone can find the sequence using suitable software.
In the limit, there are countably many compact elements of  $D_\infty$. The classical cardinality of $D_\infty$ is that of $P{\mathbb N}$.
Then $D_\infty\cong[D_\infty\to D_\infty]$, meaning the domain of functions that preserve directed joins.
Dana Scott discovered this and the "$P\omega$" model of the untyped $\lambda$-calculus after previously believing there was no "mathematical" model of it.  (Of course he knew from Church–Rosser that it is syntactically consistent.)
See Scott on the consistency of the lambda calculus for further discussion of that history.
I'm struggling to find where Scott first introduced the $D_\infty$ model, but there is a paragraph about it in
An Outline of a Mathematical Theory of Computation.
I removed the "cardinality" tags from this question because they are misleading, cf. @Wojowu's (1 2) and @ToddTrimble's (1) comments above.
