Variations on $\Delta^1_1$ Determinacy and $\aleph_{\omega_1}$

TL;DR

The paper explores a weaker form of $ ext{Delta}^1_1$ Turing determinacy, showing it implies the existence of models with various infinite cardinals and connecting it to classical determinacy results.

Contribution

It introduces a weaker determinacy principle and demonstrates its implications for the existence of models with infinite cardinals, improving known results on Borel determinacy.

Findings

Weak-Turing-Det}_ ho ( ext{Delta}^1_1)$ implies models with all $ ext{aleph}_ u$ for $ u< ext{omega}_1^{ ext{CK}}$

If every cofinal $ ext{Delta}^1_1$ set contains a degree and its jump, then models with all $ ext{aleph}_ u$ exist

Weak-Turing-Det}_ ho ( ext{Delta}^1_1)$ implies $ ext{Delta}^1_1$ determinacy

Abstract

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 $\Sigma^1_1$.