VARIATIONS ON DETERMINACY AND

We consider a seemingly weaker form of $\Delta^1_1$ Turing determinacy. Let $2 \leq \rho < \omega_1^{\textrm{CK}}$, $\textrm{Weak-Turing-Det}_\rho (\Delta^1_1)$ is the statement: Every set reals cofinal in degrees contains two distinct, $\Delta^0_\rho$-equivalent reals. show $\textrm{ZF}^-$: implies: for every $\nu \omega_1^{\textrm{CK}}$ there transitive model: $M \models \textrm{ZF}^- + \aleph_\nu \textrm{ exists}$. As corollary: If both degree and its jump, then -- With simple proof, this improves upon well-known result Harvey Friedman on strength Borel determinacy (though not assessed level-by-level). Invoking Tony Martin's proof determinacy, implies further that imparts weak properties to class $\Sigma^1_1$.