For the last point, besides texts explicitly on computability-theoretic descriptive set theory (e.g. the hard-to-find Mansfield-Weitkampf, the freely-accessible section $3$ of Moschovakis' book, or these brief notes of Hjorth) I think the literature on **represented spaces** will be useful. There is a lot written about this; I recommend this recent survey of Schroeder (and the sources in its bibliography). Granted, this is somewhat more geared towards *computable analysis* than *(effective) descriptive set theory*, but there is still strong overlap.

Roughly speaking, suppose we have a topological space $\mathcal{X}$ equipped with a canonical basis. Each point in $\mathcal{X}$ can be described by a set of basis elements, and examples like $\mathbb{R}, 2^\mathbb{N}$, and $\mathbb{N}^\mathbb{N}$ suggest that with "naturally-occuring" spaces it is often the case that such representations yield a good computability theory (we have notions of "computable elements" of each of those spaces). Polish spaces arise as a natural generalization of this setting; note that completeness can be thought of as the analogue of "every infinite sequence of decimal digits corresponds to a real number." Of course the situation is a bit more complicated than this, as the theory of represented spaces indicates, but this is still a good starting point.

Moreover, since all "nontrivial" Polish spaces are *Borel-isomorphic*, we get a general independence principle for "high-level" descriptive set theoretic questions. This helps motivate them on the classical side.

However, I **don't** really think that (set-theoretic) forcing and inner models are best approached via computability theory, even for a computability theorist; while they do have computability-theoretic aspects, focusing too much on this will obscure the fundamentally set-theoretic nature of the topic which is necessary to any deep understanding. In particular, while there is a notion of forcing in classical computability theory, in my opinion it's far too weak to yield a good bridge to set-theoretic forcing.

I think to a large extent there's nothing to do but grab the bull by the horns. Certainly not all expositions of forcing are equally clear, but in my experience ones which attempt a significantly simplified presentation (e.g. this "analytic" approach of Scott) don't really serve well as introductions to the *fully general* theory. The two standard approaches to forcing are via *posets* and *Boolean-valued models*, and I would just pick one and study it directly. In my experience the poset-based approach is far simpler, and it's what I learned; however, the Boolean-valued approach avoids any need to talk about generic objects directly, which may be a significant intuitive benefit if one is uncomfortable with model theory.

For general set-theoretic forcing and inner models, I think the right starting point *(overlooking the concern of the paragraph above)* is **hyperarithmetic theory**. Not only is this a "lightface" analogue of the Borel hierarchy (so it's relevant to the previous point too), it is tightly connected to the construction of $L$: in a very strong sense, $L$ is what you get when you "continue the hyperarithmetic hierarchy through the ordinals" (the relevant term here is "**master codes**" - see here).

Hyperarithmetic theory is also useful for understanding set-theoretic forcing. Forcing in *classical* computability theory (e.g. Jockusch-Soare forcing) is a bit too simple to provide a good starting point for set-theoretic forcing in my opinion. One wants to shift from classical computability theoretic forcing to forcing over some "set theory flavored" structure. It turns out that **forcing over ***levels of $L$* is a very good context for doing this. For example, consider the proof that a comeager set of reals $x$ satisfy $\omega_1^{CK}(x)=\omega_1^{CK}$; this is basically "Cohen forcing over $L_{\omega_1^{CK}}$." In general, there's a rich theory of **forcing over admissible sets**.

My personal favorite treatment of the hyperarithmetic hierarchy and related notions is (the first few parts of) Sacks' book **Higher recursion theory** *(it should be available via ProjectEuclid, but that link doesn't work at the moment)*.

*Now granted, in a precise sense we can make sense of forcing "computably" rather than "hyperarithmetically;" however, while it's a very nice result I don't think this is a good way to approach the subject as a beginner from computability theory.*

One final topic worth mentioning re: forcing, specifically connected with combinatorial topics like $\mathsf{CH}$, is the theory of **effective cardinal characteristics of the continuum**. This amounts to a "computable analogue" of the analysis of the interval $[\aleph_1,2^{\aleph_0}]$ via forcing, with set-theoretic forcing notions usually having corresponding computability-theoretic ones which yield analogous results. For example, we have a direct connection between the following two facts:

It is (relatively) consistent with $\mathsf{ZFC}$ that there is a cardinal $\kappa$ such that $(i)$ there is a set $F$ of functions $\mathbb{N}\rightarrow\mathbb{N}$ such that every $g:\mathbb{N}\rightarrow\mathbb{N}$ is *escaped* by some $f\in F$, but $(ii)$ there is a set $F$ of functions $\mathbb{N}\rightarrow\mathbb{N}$ such that every $g:\mathbb{N}\rightarrow\mathbb{N}$ is *dominated* by some $f\in F$.

There are hyperimmune degrees *(= compute a function escaping all computable functions)* which are not high *(= compute a function dominating all computable functions)*.

**Note that a ***consistency result* is replaced by an *outright fact*.

Here the idea is to shift from **cardinality** on the set-theoretic side to **lowness/highness notions** on the computability-theoretic side. Moreover, while the applications generally go from the set-theoretic version to the computability-theoretic version, there is at least one case where we get a result on the set-theoretic side building on an initial result on the computability-theoretic side; the set-theoretic result (which was much harder) was gotten by Ivan Ongay-Valverde, and the computability-theoretic result is joint between Ivan and myself.

However, I think that this is a bit of a double-edged sword: while much more accessible to the classical computability theorist than hyperarithmetic theory, the conceptual shift in the previous sentence does lose a lot of the set-theoretic "spirit." I'm not sure how much I recommend it as a pedagogical starting point.

Computability Theoryby S. Barry Cooper. (Another person who has written about the connection between computability and forcing is Jan Krajíček, but to be honest, I find him hard to read.) $\endgroup$ – Timothy Chow Apr 7 at 15:45e.g.a countable ordinal is both a particular kind of poset structure imposed on $\mathbb{N}$ also a specific kind of set ordered by $\in$. The absoluteness of ordinals vs. non-absoluteness of cardinals is also a good, very basic, example of how to use the Lévy hierarchy. $\endgroup$ – Robert Furber Apr 8 at 1:43It is incomprehensible near1)"where I can descriptive set theory"(e.g., is a word missing?)What is abbr. "CH"? Computability hierarchy? Is "CH" a typo (e.g., of "CT" (computability theory))? $\endgroup$ – Peter Mortensen Apr 8 at 8:502)1more comment