On the admissible sets of type $\mathbb{H}\mathbb{Y}\mathbb{P}(\mathfrak{M})$ over recursively saturated models

Abstract

Some effective expression is obtained for the elements of an admissible set $\mathbb{H}\mathbb{Y}\mathbb{P}(\mathfrak{M})$ as template sets. We prove the Σ-reducibility of $\mathbb{H}\mathbb{Y}\mathbb{P}(\mathfrak{M})$ to $\mathbb{H}\mathbb{F}(\mathfrak{M})$ for each recursively saturated model $\mathfrak{M}$ of a regular theory, give a criterion for uniformization in $\mathbb{H}\mathbb{Y}\mathbb{P}(\mathfrak{M})$ for each recursively saturated model $\mathfrak{M}$, and establish uniformization in $\mathbb{H}\mathbb{Y}\mathbb{P}(\mathfrak{N})$ and $\mathbb{H}\mathbb{Y}\mathbb{P}(\Re ')$, where $\mathfrak{N}$ and $\Re '$ are recursively saturated models of arithmetic and real closed fields. We also prove the absence of uniformization in $\mathbb{H}\mathbb{F}(\mathfrak{M})$ and $\mathbb{H}\mathbb{Y}\mathbb{P}(\mathfrak{M})$ for each countably saturated model $\mathfrak{M}$ 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 Σ-definable Skolem functions, but lacking definable Skolem functions in every extension by finitely many constants.