Extending the Cooper minimal pair theorem
In the study of cappable asd noncappable properties of the recursively enumerable (r.e.) degrees, Lempp suggested a conjecture which asserts that for all r.e. degrees a and b, if a not less than or equal to b then there exists an r.e. degree c such that c less than or equal to a and c not less than or equal to b and c is cappable. We shall prove in this paper that this conjecture holds under the condition that a is high. Working below a high r.e. degree h, we show that for any r.e. degree b with h not less than or equal to b, there exist r.e. degrees a(o) and a(1) such that a(o), a(1) not less than or equal to b, a(o), a(1) less than or equal to h, and a(o) and a(1) form a minimal pair.
minimal pair recursively enumerable degree