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We provide an elementary proof of the fixpoint alternation hierarchy
in arithmetic, which in turn allows us to simplify the proof of the
modal mu-calculus alternation hierarchy. We further show that the
alternation hierarchy on the binary tree is strict, resolving a
problem of Niwiński.
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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