How to prove representation-independent independence results

A true assertion about the input-output behavior of a Turing machine M may be independent of (i.e., impossible to prove in) a theory T because the computational behavior of M is particularly opaque, or because the function or set computed by M is inherently subtle. The latter sorts of representation-independent independence results are more satisfying. For π₂ assertions, the best-known techniques for proving independence yield representation-independent results as a matter of course. This paper illustrates current understanding of unprovability for π₂ assertions by demonstrating that very weak conditions on classes of sets S and R guarantee that there exists a set L₀ ε R - S such that L₀ is not probably infinite (hence, not provably nonregular, nondeterministics, noncontext-free, not in P, etc.). Under slightly stronger conditions, such L₀'s may be found within every L ε R - S.