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The arithmetic and turing degrees are not elementary equivalent.

Richard A. Shore — 1984

Archiv für mathematische Logik und Grundlagenforschung

The strength of Turing determinacy within second order arithmetic

Antonio Montalbán; Richard A. Shore — 2016

Fundamenta Mathematicae

We investigate the reverse mathematical strength of Turing determinacy up to Σ₅⁰, which is itself not provable in second order arithmetic.

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