At the outer bounds of computational complexity classes are those defined through computability theory (AKA recursion theory). This is where we get the well known complexity classes such as R, RE, and co-RE. However, something else defined through computability theory is the concept of an immune set. Immune sets are ones where you are unable to enumerate even an infinite subset.
My question is this: is there a complexity class of decision problems where an infinite subset of "yes" instances can be enumerated, and is there a "better" name for it than "not immune"? I ask this because complexity classes are typically named for what they are, but "immune" describes what the decision problem is not.