VARIATIONS ON $\Delta ^{1}_{1}$ DETERMINACY AND ℵ_ω1

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 $\mathcal {D}$ .