17. C. T. Chong and Liang Yu, A$\Pi^1_1$ uniformization principle ...

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6 C. T. CHONG AND LIANG YU For (ii): If y ∈ A then y = G(α) for some α < ω1. By the definition of G, there is a real x ∈ L G(α) such that x codes the set {G(β)|β < α} and P(x, G(α)) holds. But obviously β < α if ω1 and only if G(β)
A Π 1 1-UNIFORMIZATION PRINCIPLE FOR REALS 7 4. Maximal chains and thin maximal antichains in the Turing degrees In this section, we apply the uniformization principle I to solve the existence problem of Π 1 1 maximal chains and thin Π 1 1 maximal antichains in the Turing degrees. These problems were studied in [5] and [6] using direct constructions involving the Boolos-Putnam analysis of constructible reals. Here we derive the results from the principle I. Theorem 4.1 (Chong and Yu [5]). Assume (ω1) L = ω1. There is a Π 1 1 maximal chain in the Turing degrees. Proof. Define a binary relation P as follows: P(x, y) ⇔ y is a minimal cover of {(x)n|n ∈ ω}. It was shown in [5] that P is a ∆ 1 1 cofinally progressive relation (in the language of the current paper). Hence by Theorem 3.4, there is a Π 1 1 set A as prescribed. I(ii) implies that A is a maximal chain. In the case of a maximal antichain, a diagonal argument shows that under ZF C there is a thin maximal antichain in the Turing degrees. Hence the definability of a thin set with such a property becomes particularly interesting. Since every set of reals whose degrees form a maximal antichain has size 2 ℵ0 , it cannot be Σ 1 1 (else it would contain a perfect subset). The next level of definability is then Π 1 1. We apply Theorem 3.4 to construct a thin Π 1 1 maximal antichain in the Turing degrees. Theorem 4.2. Assume (2 ω ) L = 2 ω . There exists a Π 1 1 thin maximal antichain in the Turing degrees. Proof. Define a binary relation P(x, y) as follows: (1) {(x)n|n ∈ ω} 1 is not an antichain or; (2) {(x)n|n ∈ ω} ∪ {y} is an antichain and (2a) x ⊕ (y)0 ∈ L y ω1 ; (2b) {(x)n|n ∈ ω} ∪ {(y)0} is an antichain; (2c) for every z
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Page 11 and 12: (a) z ∈ A; (b) z ≥T x; (c) y =

17. C. T. Chong and Liang Yu, A$\Pi^1_1$ uniformization principle ...

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