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 $\lVert D\_n\rVert$?



What is the cardinality of Dana Scott's $D_{\infty}$?
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](https://mathoverflow.net/questions/16752/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*](https://www.researchgate.net/publication/242406560_Outline_of_a_Mathematical_Theory_of_Computation).


I removed the "cardinality" tags from this question because they are misleading, cf. @Wojowu's ([1](https://mathoverflow.net/questions/382206/what-is-the-cardinality-of-dana-scotts-d-infty#comment971620_382206) [2](https://mathoverflow.net/questions/382206/what-is-the-cardinality-of-dana-scotts-d-infty#comment971623_382206)) and @ToddTrimble's ([1](https://mathoverflow.net/questions/382206/what-is-the-cardinality-of-dana-scotts-d-infty#comment971635_382206)) comments above.


