Orbits of computably enumerable sets: low sets can avoid an upper cone

Russell Miller
2002 Annals of Pure and Applied Logic  
We investigate the orbit of a low computably enumerable (c.e.) set under automorphisms of the partial order E of c.e. sets under inclusion. Given an arbitrary low c.e. set A and an arbitrary noncomputable c.e. set C, we use the New Extension Theorem of Soare to construct an automorphism of E mapping A to a set B such that C T B. Thus, the orbit in E of the low set A cannot be contained in the upper cone above C. This complements a result of Harrington, who showed that the orbit of a
more » ... e c.e. set cannot be contained in the lower cone below any incomplete c.e. set.
doi:10.1016/s0168-0072(01)00119-1 fatcat:skffmxyfrfdclane7tt3kadsny