Variations on ${\mathrm{\Delta }}_1^1$ Determinacy and ${\mathrm{\aleph }}_{{\omega }_1}$

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