On the Hausdorff dimension of maximal chains and antichains of Turing and Hyperarithmetic degrees

Abstract: This paper investigates the Hausdorff dimension properties of chains and antichains in Turing degrees and hyperarithmetic degrees. Our main contributions are threefold: First, for antichains in hyperarithmetic degrees, we prove that every maximal antichain necessarily attains Hausdorff dimension 1. Second, regarding chains in Turing degrees, we establish the existence of a maximal chain with Hausdorff dimension 0. Furthermore, under the assumption that $\omega_1=(\omega_1)^L$, we demonstrate the existence of such maximal chains with $\Pi^1_1$ complexity. Third, we extend our investigation to maximal antichains of Turing degrees by analyzing both the packing dimension and effective Hausdorff dimension.