Presentations of Structures in Admissible Sets

We consider copies and constructivizations of structures in admissible sets. It is well known that in classical computable model theory (on natural numbers) these approaches are equivalent: a structure has computable (decidable) copy if and only if it is constructivizable (strongly constructivizable). However, in admissible sets the "if" part of this statement is not true in general. In the rst section we survey results about copies in hereditary nite superstructures and deenabil-ity (so called syntactical conditions of intrinsically computable properties). The second section is devoted to constructivizations of uncountable structures in "simplest" uncountable admissible sets. The third section contains some results on constructivizations of admissible sets within themselves. We denote by F() the set of nite rst order formulas of a signature. We also x some GG odel numbering dde : F() ! ! (d'e { GG odel number of '). In all that follows we consider only computable signatures and suppose that GG odel numberings are eeective. We also denote by F n () (n 6 !) a set of ((nite rst order) formulas of signature with no more than n alterating groups of quantiiers in prenex normal form. F 0 () is a set of quantiier-free formulas of signature. Let M be a structure of signature , A an admissible set, and let M A.