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 $\Delta^1_1$ set of reals cofinal in the Turing degrees contains two Turing distinct, $\Delta^0_\rho$-equivalent reals. We show in $\textrm{ZF}^-$: $\textrm{Weak-Turing-Det}_\rho (\Delta^1_1)$ implies: for every $\nu < \omega_1^{\textrm{CK}}$ there is a transitive model: $M \models \textrm{ZF}^- + \aleph_\nu \textrm{ exists}$. As a corollary: If every cofinal $\Delta^1_1$ set of Turing degrees contains both a degree and its jump, then for every $\nu < \omega_1^{\textrm{CK}}$, there is a transitive model: $M \models \textrm{ZF}^- + \aleph_\nu \textrm{ exists}$. -- 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, $\textrm{Weak-Turing-Det}_\rho (\Delta^1_1)$ implies $\Delta^1_1$ determinacy. We show further that $\Delta^1_1$ determinacy imparts weak determinacy properties to the class $\Sigma^1_1$.

The results in this paper were motivated by the following question of Sacks. Suppose T is a recursive theory with countably many countable models. What can you say about the least ordinal α such that all models of T have Scott rank below α? If Martin's conjecture is true for T then α ≤ ω · 2. Our goal was to look at this problem in a more abstract setting. Let E be a equivalence relation on ωω with countably many classes each of which is Borel. What can you say about the least α such that each equivalence class is ? This problem is closely related to the following question. Suppose X ⊆ ωω is and Borel. What can you say about the least α such that X is ? In §1 we answer these questions in ZFC. In §2 we give more informative answers under the added assumptions V = L or -determinacy. The final section contains related results on the separation of sets by Borel sets. Our notation is standard. The reader may consult Moschovakis [5] for undefined terms. Some of these results were proved first by Sami and rediscovered by Kechris and Marker.

Abstract In Baire space we define a sequence of equivalence relations ‹Ev ∣ v < , each Ev being with classes in + v + 1 and such that (i) Ev does not have perfectly many classes, and (ii) is countable iff < ω1. This construction can be extended cofinally in . A new proof is given of a theorem of Hausdorff on partitions of R into ω1 many sets.