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Recursion Theory by Liang Yu, Chi Tat Chong

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2.5.2 Incomparable hyperdegrees

We apply the Gandy Basis Theorem to construct a pair of incomparable hyperdegrees.

Lemma 2.5.4. If A is a image-set containing a nonhyperarithmetic real, then A is uncountable.

Proof. Suppose that A is a countable image-set containing a nonhyperarithmetic real. Then the set B = {xA | xh image} is a nonempty image-subset of A. Hence there ...

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