So I need to writeup some old results of Harrington's which imply various results about admissible ordinals. I've never really learned admissible recursion theory so what's a good reference?



Good source for admissible set theory?
I **strongly** recommend Sacks' book *[Higher recursion theory](https://projecteuclid.org/ebooks/perspectives-in-logic/Higher-Recursion-Theory/toc/pl/1235422631)* in conjunction with Barwise's *[Admissible sets and structures](https://projecteuclid.org/euclid.pl/1235418470)* (each of which is freely and legally available online via ProjectEuclid). The former is probably more oriented to what you're looking for, which sounds like $\alpha$-recursion theory specificially, but the latter is good enough that I can't not recommend it too.


Note that there is a difference between "admissible recursion theory" and "$\alpha$-recursion theory:" the $L\_\alpha$s for $\alpha$ admissible are very special types of admissible sets. A general admissible set need not support a good recursion theory at all; this was studied by Simpson, Stoltenberg-Hanssen, and others if I recall correctly, leading to terms like "resolvable admissible set" (some relevant material is in the proceedings volumes of the Oslo conferences on Generalized Recursion Theory). I suspect you're more interested in $\alpha$-recursion theory, or "near" $\alpha$-recursion theory (e.g. set generic extensions of admissible $L\_\alpha$s) than admissible recursion theory per se.


