TITLE: How (non-)computable is set theory?
QUESTION [30 upvotes]: Here is a naive outsiders perspective on set theory: A typical set-theoretical result involves constructing new models of set theory from given ones (typically with different theories for the original model and the resulting model). Typically, from a meta-perspective we are allowed (encouraged, or even required) to assume that the models are countable.
To the extent that this view is correct, set-theoretic constructions correspond to partial, multivalued operations $T : \subseteq \{0,1\}^\mathbb{N} \rightrightarrows \{0,1\}^\mathbb{N}$ which are defined on sequences coding a model of the original theory, and are outputting a sequence coding a model of the desired theory. These are multivalued, because the constructions may involve something like "pick a generic filter for this forcing notion".
Question: Are these operations typically computable, and if not, how non-computable are they?
The Weihrauch degrees (https://arxiv.org/abs/1707.03202) provide a framework for classifying non-computability of operations of these types. An answer, however, could also take forms like "For arguments like this, a resulting model is typically computable in the join of the original model and a 1-generic."

REPLY [7 votes]: For what it's worth, you might also consider the sets computable by Ordinal Turing Machines (see Koepke's paper, Turing Computations on Ordinals, arXiv:math/0502264v1 [math.LO] 13 Feb 2005).  A central theorem regarding Ordinal Turing Machines is the following:

A set $x$ of ordinals is ordinal computable from a finite set of ordinal parameters if and only it is an element of the constructible universe $L$.

Relative to this type of Turing machine, the non-computable sets are just the non-constructible sets. 
It would be interesting to study the degree theory of the non-constructible sets relative to Ordinal Turing Machines.