Computing strength of Structures Related to the field of Real numbers

The behavior of structures in generic extensions of the universe has been studied from a number of different angles; for example, Baldwin, Laskowski, and Shelah [2] studied the conditions under which non-isomorphic structures may become isomorphic, and Knight, Montalban, and Schweber [8] (and independently Kaplan and Shelah [7]) studied structures existing in every generic extension of the universe by some forcing. In the latter example, general results about such “generically presentable” structures led to a new proof of a result of Harrington saying that if T is a counterexample to Vaught’s Conjecture, then T has models of cardinality א1 with arbitrarily large Scott ranks less than ω2. (There are now at least three new proofs of this result. In addition to the one in [8], there is one by Baldwin, S.-D. Friedman, Koerwien, and Laskowski [1] and one by Larson [10]; these other proofs do not use generically presentable structures directly, but do use related ideas.) We can do more with generic extensions. In [8], the third author defined a notion that lets us compare the computing power of structures of any cardinality.