Possible degrees in recursive copies Original Research Article

Let View the MathML source be a recursive structure, and let R be a recursive relation on View the MathML source. Harizanov (1991) isolated a syntactical condition which (with additional effectiveness conditions) is necessary and sufficient for View the MathML source to have recursive copies in which the image of R is r.e. of arbitrary r.e. degree. We had conjectured that a certain extension of Harizanovʹs syntactical condition would (with some effectiveness conditions) be necessary and sufficient for View the MathML source to have recursive copies in which the image of R is ∑α0 of arbitrary ∑α0 degree, but this is not the case. Here we give examples illustrating some restrictions on the possible ∑α0 degrees. In these examples, the image of R cannot be ∑α0 of degree d unless d possesses an “α-table” (a sequence of sets in which each one is r.e. in and above the earlier ones).