Descriptive properties on admissible sets (Q619320)

The author explores the relation between various descriptive properties on admissible sets, with focus on reduction (\(\mathsf{Red}\)), existence of universal \(\Sigma\)-functions (\(\mathsf{UF}\)), existence of \(\{0,1\}\)-valued universal \(\Sigma\)-functions (\(\mathsf{UF2}\)) and separation (\(\mathsf{Sep}\)). It is shown that admissible sets behave rather badly in that almost every relation between these properties may be realized on a certain admissible set. And in most cases, such a set can be chosen as a hereditarily finite superstructure \(\mathbb{HF}(\mathfrak{M})\) over a computable model \(\mathfrak{M}\). The paper is divided into four parts. In Part I, the relations between the following properties are stated: enumerability, uniformization, reduction, (\(\mathsf{UF}\)), (\(\mathsf{UF2}\)), separation, and extension. Part II focuses on (\(\mathsf{Red}\)): various examples of classical structures \(\mathcal{M}\) are given so that \(\mathbb{HF}(\mathfrak{M})\) does not satisfy the reduction property. Part III considers (\(\mathsf{UF}\)) and provides computable models \(\mathbb{HF}(\mathfrak{M})\) that disprove the implications \((\mathsf{UF})\to(\mathsf{Red})\) and \((\mathsf{Red})\to(\mathsf{UF})\), respectively. Finally, Part IV deals with (\(\mathsf{UF2}\)): computable structures of the form \(\mathbb{HF}(\mathfrak{M})\) are specified where (\(\mathsf{Sep}\)) and (\(\mathsf{UF2}\)) fail, and where (\(\mathsf{Sep}\)) and (\(\mathsf{UF}\)) fail, but (\(\mathsf{UF2}\)) still holds.