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Isomorphism Types of Rogers Semilattices in the Analytical Hierarchy

    https://doi.org/10.1142/9789811278631_0004Cited by:0 (Source: Crossref)
    Abstract:

    A numbering of a countable family S is a surjective map from the set of natural numbers ω onto S. A numbering ν is reducible to a numbering µ if there is an effective procedure which given a ν-index of an object from S, computes a µ-index for the same object. The reducibility between numberings gives rise to a class of upper semilattices, which are usually called Rogers semilattices. This chapter studies Rogers semilattices for families SP (ω) belonging to various levels of the analytical hierarchy. We prove that for any non-zero natural numbers mn, any non-trivial Rogers semilattice of a Πm1-computable family cannot be isomorphic to a Rogers semilattice of a Πn1-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.