An interesting feature of domain theory is to use partial orders in order to provide a mathematical model for the *computational approximation* in a potentially infinite computational process (e.g. computing all digits of $\pi$) that lies beyond the power of our finite-resource computers.

Roughly speaking, from a domain-theoretic perspective, the *pieces of information* in a computational process form a partial order. Under certain conditions, an infinite set of *compatible* pieces of partial information may *converge* to the *ultimate answer* beyond the computer's computational power.

So the computer will never know the final answer (e.g. all digits of $\pi$) but it deals with its *approximations*. For instance, at a certain point it finds out that $\pi\in I=[3.14, 3.15]$ and after a few more steps of computing it comes up with the fact that $\pi\in J=[3.141, 3.142]$. So the interval $J$ can be viewed as a piece of information that *extends* its predecessor $I$ via inverse inclusion order, $J\subseteq I$.

This approach may sound quite familiar to the set theorists as it reminds the way they deal with *conditions* in a *forcing notion*. The ultimate object is the *generic filter* (and whatever made out of it) which lives in the forcing extension but not in the ground model. However, people living in the ground model have a degree of access to such a "transcendental" object through its *approximations* that exist in their world.

Based on the presented analogy, one may expect forcing and domain theory to share some similar concepts and techniques. But my search didn't reveal that much along these lines, except a master thesis of Håkon Briseid (under the supervision of Dag Normann), titled "*Generic Functions in Scott Domains*". Its abstract reads as:

In this thesis we will apply forcing to domain theory. When a Scott domain represents a function space, each function will be a filter in the basis of the domain. By using partially ordered basis as the forcing relation, each generic filter $G$ yields a model of $ZFC$ in which $G$ is a function, given some other model of $ZFC$ containing this basis. Such generic functions are the main concern of this thesis. ...

My question is about the existence of possibly deeper connections between these two branches of mathematical logic.

Question.What are other examples of papers and theses relating set theory and domain theory through applying forcing in domain theoretic theorems and constructions?

`domain-theory`

tag to add to the original post. Instead, I used the`order-theory`

tag. However, as domain theory is a vast and active branch of mathematical logic, I think it really deserves an independent tag here on MathOverflow. $\endgroup$ – Morteza Azad Jun 22 '18 at 14:44Connections between Complexity Theory & Set Theory$\endgroup$ – Morteza Azad Jun 25 '18 at 8:36