Inseparability in recursive copies

Abstract

In [7] and [8], it is established that given any abstract countable structure S and a relation R on S, then as long as S has a recursive copy satisfying extra decidability conditions, R will be ∑⁰_α on every recursive copy of S iff R is definable in $\text{L}$S by a special type of infinitary formula, a ∑^r_α(p̄) formula. We generalize the typ e of constructions of these papers to produce conditions under which, given two disjoint relations R₁ and R₂ on S, there is a recursive copy of S in which R₁ and R₂ are

⁰_α inseparable. We then apply these theorems to specific everyday structures such as linear orderings, boolean algebras andvector spaces.