Variations on $Δ^1_1$ Determinacy and $\aleph_{ω_1}$

Abstract

We consider a seemingly weaker form of $Δ^1_1$ Turing determinacy. Let $2 \leq ρ< ω_1^{\textrm{CK}}$, $\textrm{Weak-Turing-Det}_ρ(Δ^1_1)$ is the statement: Every $Δ^1_1$ set of reals cofinal in the Turing degrees contains two Turing distinct, $Δ^0_ρ$-equivalent reals. We show in $\textrm{ZF}^-$: $\textrm{Weak-Turing-Det}_ρ(Δ^1_1)$ implies: for every $ν< ω_1^{\textrm{CK}}$ there is a transitive model: $M \models \textrm{ZF}^- + \aleph_ν\textrm{ exists}$. As a corollary: If every cofinal $Δ^1_1$ set of Turing degrees contains both a degree and its jump, then for every $ν< ω_1^{\textrm{CK}}$, there is a transitive model: $M \models \textrm{ZF}^- + \aleph_ν\textrm{ exists}$. -- 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, $\textrm{Weak-Turing-Det}_ρ(Δ^1_1)$ implies $Δ^1_1$ determinacy. We show further that $Δ^1_1$ determinacy imparts weak determinacy properties to the class $Σ^1_1$.