Rogers semilattices in the analytical hierarchy: The case of finite families

Abstract

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 $E_n^1$ we denote ${\Pi }_n^1$ if $n$ is odd, and ${\Sigma }_n^1$ if $n$ is even. We show that for a finite family $S$ of $E_n^1$ sets, its Rogers $E_n^1$-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 $E_n^1$-semilattice is upwards dense.

26 pages