Possible degrees in recursive copies
C. J. Ash, Julia F. Knight · Annals of Pure and Applied Logic · 1995
Let A be a recursive structure, and let R be a recursive relation on A. Harizanov (1991) isolated a syntactical condition which (with additional effectiveness conditions) is necessary and sufficient for A 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 Harizanov's syntactical condition would (with some effectiveness conditions) be necessary and sufficient for A 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).