\(\Sigma_ 2\) induction and infinite injury priority arguments. II. Tame \(\Sigma_ 2\) coding and the jump operator (Q1368581)

The authors consider models of the base theory \(PA^-\) that satisfy \(\Sigma_2\)-collection but not \(\Sigma_2\)-induction. They have shown elsewhere that such a model can contain no incomplete high r.e. degree. In this paper they prove that every such model must satisfy one of the following two conditions and that each condition is satisfied by some model: (a) all incomplete r.e. degrees are low; (b) there are exactly three degrees that are jumps of r.e. degrees. The proof proceeds by blocking for \(\Pi_2\) cuts.