From the hypothesis that all Turing closed games are determined we prove: (1) the Continuum Hypothesis and (2) every subset of ℵ1 is constructible from a real.

In Baire space N = ω ω we define a sequence of equivalence relations $\langle E_\nu| \nu , each E v being Σ 1 1 with classes in Π 0 1 + ν + 1 and such that (i) E ν does not have perfectly many classes, and (ii) N/E ν is countable iff $\omega^L_\nu . This construction can be extended cofinally in (δ 1 3 ) L . A new proof is given of a theorem of Hausdorff on partitions (...) of R into ω 1 many Π 0 3 sets. (shrink)

We consider a seemingly weaker form of $\Delta ^{1}_{1}$ Turing determinacy.Let $2 \leqslant \rho < \omega _{1}^{\mathsf {CK}}$, $\textrm{Weak-Turing-Det}_{\rho }$ 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 $\mathsf {ZF}^-$ : $\textrm{Weak-Turing-Det}_{\rho }$ implies: for every $\nu < \omega _{1}^{\mathsf {CK}}$ there is a transitive model ${M \models \mathsf {ZF}^{-} + \textrm{``}\aleph _{\nu } \textrm{ exists''.}}$ As a corollary:If every cofinal $\Delta ^{1}_{1}$ set of (...) Turing degrees contains both a degree and its jump, then for every $\nu < \omega_1^{\mathsf{CK}}$, there is atransitive model: $M \models \mathsf{ZF}^{-} + \textrm{``}\aleph_\nu \textrm{ exists''.}$ •With a simple proof, this improves upon a well-known result of Harvey Friedman on the strength of Borel determinacy.•Invoking Tony Martin’s proof of Borel determinacy, $\textrm{Weak-Turing-Det}_{\rho }$ implies $\Delta ^{1}_{1}$ determinacy.•We show further that, assuming $\Delta ^{1}_{1}$ Turing determinacy, or Borel Turing determinacy, as needed:–Every cofinal $\Sigma ^{1}_{1}$ set of Turing degrees contains a “hyp-Turing cone”: ${\{x \in \mathcal {D} \mid d_{0} \leqslant _{T} x \leqslant _{h} d_{0} \}}$.–For a sequence $_{k < \omega }$ of analytic sets of Turing degrees, cofinal in $\mathcal {D}$, $\bigcap _{k} A_{k}$ is cofinal in $\mathcal {D}$. (shrink)