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 2 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 2 assertions by demonstrating that very weak conditions on classses of sets S and R guarantee that there exists a set L 0 2 R ? S such that L 0 is not provably innnite (hence, not provably nonregular, nondeterministic, non-context-free, not in P, etc.). Under slightly stronger conditions, such L 0 s may be found within every L 2 R ? S.