VARIATIONS ON 
$\Delta ^{1}_{1}$
 DETERMINACY AND ℵ ω 1
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 show in 
$\mathsf {ZF}^-$
 : 
$\textrm{Weak-Turing-Det}_{\rho }(\Delta ^{1}_{1})$
 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 (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, 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 
$(A_{k})_{k < \omega }$
 of analytic sets of Turing degrees, cofinal in 
$\mathcal {D}$
 , 
$\bigcap _{k} A_{k}$
 is cofinal in 
$\mathcal {D}$
 .