It seems to me that there are scattered references of deep relationships between descriptive set theory and computability theory. For one, the relationship between the [Borel hierarchy](https://en.wikipedia.org/wiki/Descriptive_set_theory#Borel_hierarchy) and the [Polynomial hierarchy](https://en.wikipedia.org/wiki/Polynomial_hierarchy) for one, and the relationship between [topology and computability](https://www.sciencedirect.com/science/article/pii/S1571066104051357).

I would like to understand things such as:
- What is forcing, and how does one use it to prove independence of CH?
- What precisely is a [inner model](https://en.wikipedia.org/wiki/Inner_model)?
- [Why are polish spaces an important setting for descriptive set theory](https://en.wikipedia.org/wiki/Polish_space)?

It would be fantastic if these could be answered from the perspective of _computability_. For example, I know that the Cantor set has [good computability properties](http://math.andrej.com/2007/09/28/seemingly-impossible-functional-programs/), and I (possibly very incorrect) believe that this is related to descriptive set theory.

I'd greatly appreciate references to lecture notes, talks, video lectures, textbooks. Pretty much anything where I can descriptive set theory with a computational bent would be great.