Example: The Truth Set of Arithmetic
Every arithmetical set is hyperarithmetical, but there are many other hyperarithmetical sets. One example of a hyperarithmetical, nonarithmetical set is the set T of Gödel numbers of formulas of Peano arithmetic that are true in the standard natural numbers . The set T is Turing equivalent to the set, and so is not high in the hyperarithmetical hierarchy, although it is not arithmetically definable by Tarski's indefinability theorem.
Read more about this topic: Hyperarithmetical Theory
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