∑2 Induction and infinite injury priority arguments, part III: Prompt sets, minimal pairs and Shoenfield’s Conjecture

摘要

We prove that in everyB∑ 2 model (one satisfies ∑2 collection axioms but not ∑2 induction), every recursively enumerable (r.e.) set is either prompt or recursive. Consequently, over the base theory ∑2 collection, the existence of r.e. minimal pairs is equivalent to ∑2 induction. We also refute Shoenfield’s Conjecture inB∑ 2 models.