A numbering of a countable family $S$ is a surjective map from the set of natural numbers $omega$ onto $S$. A numbering $ u$ is reducible to a numbering $mu$ if there is an effective procedure which given a $ u$-index of an object from $S$, computes a $mu$-index for the same object. The reducibility between numberings gives rise to a class of upper semilattices, which are usually called Rogers semilattices. The paper studies Rogers semilattices for families $S subset P(omega)$ belonging to various levels of the analytical hierarchy. We prove that for any non-zero natural numbers $m eq n$, any non-trivial Rogers semilattice of a $Pi^1_m$-computable family cannot be isomorphic to a Rogers semilattice of a $Pi^1_n$-computable family. One of the key ingredients of the proof is an application of the result by Downey and Knight on degree spectra of linear orders.
Download