  We study computable PAC (CPAC) learning as introduced by Agarwal et al.
(2020). First, we consider the main open question of finding characterizations
of proper and improper CPAC learning. We give a characterization of a closely
related notion of strong CPAC learning, and provide a negative answer to the
COLT open problem posed by Agarwal et al. (2021) whether all decidably
representable VC classes are improperly CPAC learnable. Second, we consider
undecidability of (computable) PAC learnability. We give a simple general
argument to exhibit such ndecidability, and initiate a study of the
arithmetical complexity of learnability. We briefly discuss the relation to the
undecidability result of Ben-David et al. (2019), that motivated the work of
Agarwal et al.
