Logic Seminar

Let L be a countable language, let I be an isomorphism-type of countable L-structures and let a be an oracle. We say that I is "a-strange" if it contains a recursive-in-a structure and its Scott rank is exactly omega_1^a. Such structures exist but there are no known natural examples. Theorem(AD): If C is a collection of aleph_1 isomorphism-types of countable structures, then for a Turing cone of a's, no member of C is a-strange.