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- W3119868491 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 $Sigma^1_1$." @default.
- W3119868491 created "2021-01-18" @default.
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- W3119868491 date "2021-01-07" @default.
- W3119868491 modified "2023-09-27" @default.
- W3119868491 title "VARIATIONS ON DETERMINACY AND" @default.
- W3119868491 doi "https://doi.org/10.1017/jsl.2020.47" @default.
- W3119868491 hasPublicationYear "2021" @default.
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