VARIATIONS ON DETERMINACY AND <i>ℵ</i><sub><i>ω</i><sub>1</sub></sub>
VARIATIONS ON DETERMINACY AND <i>ℵ</i><sub><i>ω</i><sub>1</sub></sub>
Abstract We consider a seemingly weaker form of $\Delta ^{1}_{1}$ Turing determinacy. Let $2 \leqslant \rho < \omega _{1}^{\mathsf {CK}}$ , $\textrm{Weak-Turing-Det}_{\rho }(\Delta ^{1}_{1})$ is the statement: Every $\Delta ^{1}_{1}$ set of reals cofinal in the Turing degrees contains two Turing distinct , $\Delta ^{0}_{\rho }$ - equivalent reals. We …