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Let $\Sigma_n$ and $\Pi_n$ be two levels of the levy hierarchy. We consider the set of formulas $$\Gamma = \left\{ \phi \wedge \psi, \phi \in \Sigma_n, \psi \in \Pi_n \right\}$$

Is there a common name for such a $\Gamma$ ? Something like $\Sigma_n \cup \Pi_n$ or $\Sigma_n \wedge \Pi_n$ ?

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  • $\begingroup$ I doubt there's a standard notation; using $\sum_n\land\Gamma_n$ (with an explanation) seems reasonable $\endgroup$ May 18 at 12:16

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The standard notation is $\Sigma_n\wedge\Pi_n$. See for example the top of page $23$ in Arnie Miller's paper On the Borel Classification of the Isomorphism Class of a Countable Model.

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