ON THE ADMISSIBLE SETS OF TYPE HYP(M) OVER RECURSIVELY SATURATED MODELS

Some effective expression is obtained for the elements of an admissible set HYP(M) as template sets. We prove the S-reducibility of HYP(M) to HF(M) for each recursively saturated model M of a regular theory, give a criterion for uniformization in HYP(M) for each recursively saturated model M, and establish uniformization in HYP(N) and HYP(R'), where N and R' are recursively saturated models of arithmetic and real closed fields. We also prove the absence of uniformization in HF(M) and HYP(M) for each countably saturated model M of an uncountably categorical theory, and give an example of this type of theory with definable Skolem functions. Furthermore, some example is given of a model of a regular theory with S-definable Skolem functions, but lacking definable Skolem functions in every extension by finitely many constants.