Δ2 constructions are the simplest priority constructions in which Player I controls a set W. Such constructions take place on three trees, T0, T1, and T2. In a typical situation, W is non-computable and the success of the construction requires the entry of numbers into W at a stage at which such numbers are useful to the construction. To show that this must happen, the construction will try to compute more and more of W at prescribed sets of stages, so as W is not computable, these attempts must fail; and the failure will produce numbers that enter W when they are useful.

We discuss general Δ2 constructions in Section 4.1, stating a lemma describing relationships between nodes of that level. The lemma is applied to prove an upward cone avoidance theorem in Section 4.2, the Sacks Splitting Theorem in Section 4.3, basic facts about backtracking are presented in Section 4.4 and a permitting construction is carried out in Section 4.5.

The Δ2 Level

Unlike Σ1 constructions, Δ2 constructions will have level 2 requirements. However, these requirements will be of a very special nature, making these constructions somewhat similar to those at the Σ1 level. Prior to the existence of the classification of priority arguments using the arithmetical hierarchy, a differentiation was made between finite injury and infinite injury priority constructions only, and Δ2 constructions were classified as finite injury. But even at that time, a distinction was made between the two types of finite injury constructions.