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