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 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 R 1 and R 2 on S, there is a recursive copy of S in which R 1 and R 2 are ▪ 0 α inseparable. We then apply these theorems to specific everyday structures such as linear orderings, boolean algebras andvector spaces.
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