A numbering of a countable family $S$ is a surjective map from the set of natural numbers $\omega$ onto $S$. The paper studies Rogers semilattices, i.e. upper semilattices induced by the reducibility between numberings, for families $S\subset P(\omega)$. Working in set theory ZF+DC+PD, we obtain the following results on families from various levels of the analytical hierarchy. For a non-zero number $n$, by $E^1_n$ we denote $\Pi^1_n$ if $n$ is odd, and $\Sigma^1_n$ if $n$ is even. We show that for a finite family $S$ of $E^1_n$ sets, its Rogers $E^1_n$-semilattice has the greatest element if and only if $S$ contains the least element under set-theoretic inclusion. Furthermore, if $S$ does not have the $\subseteq$-least element, then the corresponding Rogers $E^1_n$-semilattice is upwards dense.
|Title of host publication||ALC2019 Procceding|
|Publication status||Submitted - Oct 2 2020|