Good source for admissible set theory?

Question

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?

Answer 1

I strongly recommend Sacks' book Higher recursion theory in conjunction with Barwise's Admissible sets and structures (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.