Dual Easy Uniformization and Model-Theoretic Descriptive Set Theory

Abstract

It is well known that, in the terminology of Moschovakis, Descriptive set theory (1980), every adequate normed pointclass closed under $\forall^\omega$ has an effective version of the generalized reduction property (GRP) called the easy uniformization property (EUP). We prove a dual result: every adequate normed pointclass closed under $\exists^\omega$ has the EUP. Moschovakis was concerned with the descriptive set theory of subsets of Polish topological spaces. We set up a general framework for parts of descriptive set theory and prove results that have as special cases not only the just-mentioned topological results, but also corresponding results concerning the descriptive set theory of classes of structures. Vaught (1973) asked whether the class of $\mathrm{cPC}_\delta$ classes of countable structures has the GRP. It does. A $cPC(A)$ class is the class of all models of a sentence of the form $\neg\exists\bar{K}\phi$, where $\phi$ is a sentence of $\mathscr{L}_{\infty\omega}$ that is in $A$ and $\bar{K}$ is a set of relation symbols that is in $A$. Vaught also asked whether there is any primitive recursively closed set $A$ such that some effective version of the GRP holds for the class of $\mathrm{cPC}(A)$ classes of countable structures. There is: The class of $\mathrm{cPC}(A)$ classes of countable structures has the EUP if $\omega \in A$ and $A$ is countable and primitive recursively closed. Those results and some extensions are obtained by first showing that the relevant classes of classes of structures, which Vaught showed normed, are in a suitable sense adequate and closed under $\exists^\omega$, and then applying the dual easy uniformization theorem.