The family of all recursively enumerable classes of finite sets
Thomas G. McLaughlin · Transactions of the American Mathematical Society · 1971
We prove that if $P(x)$ is any first-order arithmetical predicate which enumerates the family Fin of all r.e. classes of finite sets, then $P(x)$ must reside in a level of the Kleene hierarchy at least as high as $\prod _3^0 - \Sigma _3^0$. (It is more easily established that some of the predicates $P(x)$ which enumerate Fin do lie in $\prod _3^0 - \Sigma _3^0$.)