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Is it possible to realize arithmetic hierarchy in algebraic number theory?

For example, consider a $\Pi^0_4$ statement of the form $\forall x \exists y \forall z \exists w \phi(x,y,z,w)=0$ where $\phi$ is some recursive function and $x,y,z,w$ are elements of an algebraic number field. Are there such statements in algebraic number theory?

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logics restricted in arithmetic hierarchy

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  • $\begingroup$ Having $\phi$ be a recursive function (i.e. computable function) is rather like having another existential quantifier, to say that it converges, and so it may be more natural/clear to strip that out and say that $\phi$ is primitive recursive, since the extra existential quantifer can be absorbed into $\exists w$. $\endgroup$ Commented Aug 8, 2024 at 21:17

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