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.