Abstract

A minimal extension of a Π01 class P is a Π01 class Q such that P ⊂ Q, Q – P is infinite, and for any Π01 class R, if P ⊂ R ⊂ Q, then either R – P is finite or Q – R is finite; Q is a nontrivial minimal extension of P if in addition P and Q′ have the same Cantor-Bendixson derivative. We show that for any class P which has a single limit point A, and that point of degree ≤ 0, P admits a nontrivial minimal extension. We also show that as long as P is infinite, then P does not admit any decidable nontrivial minimal extension Q