Modern reference on PA degrees?

Question

I'm currently trying to work my way around some papers from Jockush et al, and PA degrees come up frequently. I'd be interested in a modern reference/survey summarizing the main results on the subject, if such a thing exists. My usual references don't seem to discuss PA degrees.

Answer 1

If you are just looking to understand the basic results for $PA$ degrees I would recommend "Reverse Mathematics: Problems, Reductions and Proofs" by Mummert and Dzhafarov. In particular look at section 2.8 "Trees and PA degrees" which goes over some of the classical results about $PA$ degrees and basis theorems.

I would also highly recommend "Turing computability: Theory and Applications" Section 10 will go over similar results and I believe there is a nicer proof of the fact that a set has $PA$ degree if and only if it computes a diagonally non recursive function. It also has a proof that a set with $PA$ degree computes a completion of $PA$. It also has a few basis/non basis theorems which you cannot find on the reverse mathematics book.

Answer 2

Another good source is Diamondstone/Dzhafarov/Soare's survey paper $\Pi^0_1$ Classes, Peano Arithmetic, Randomness, and Computable Domination. In particular, they go into detail about weak basis theorems; while these theorems are admittedly superceded by the low basis theorem (and other "strong" basis theorems), at least in the case of the Kreisel-Shoenfield basis theorem ("Every nonempty $\Pi^0_1$ class has an element $<_T0'$") I think it's still worth seeing the result if only for "flavor."