Constructive and Non-Constructive Infinite Formulas in Computable Structures

A transition from arbitrary $L_{\omega_1\omega}$-formulas to computable formulas in the class of computable structures is considered. It is shown that transition of a certain type is possible which doubles the complexity of the formulas. In addition, the complexity jump is analyzed for the transition from an arbitrary Scott family consisting of $L_{\omega_1\omega}$-formulas to a computable Scott family in a fixed computable structure. Exact estimates of this jump are found.