Turing computable embeddings

In [3], two different effective versions of Borel embedding are defined. The first. called computable embedding, is based on uniform enumeration reducibility, while the second, called Turing computable embedding, is based on uniform Turing reducibility While [3] focused mainly on computable embeddings, the present paper considers Turing computable embeddings. Although the two notions are not equivalent, we can show that they behave alike on the mathematically interesting classes chosen for investigation in [3]. We give a "Pull-back Theorem", saying that if Phi is a Turing computable embedding of K into K-1, en for any computable infinitary sentence p in the language of K', we can find a computable infinitary sentence phi* in the language of K such that for all A is an element of K, A satisfies phi* iff Phi(A) satisfies phi, and phi* has the same "complexity" as phi (i.e., if phi is computable Sigma(alpha) or computable Pi(alpha), for alpha >= 1, then so is phi*). The Pull-back Theorem is useful in proving non-embeddability, and it has other applications as well.