Algebraicity and Implicit Definability in Set Theory

Abstract

We analyze the effect of replacing several natural uses of definability in set theory by the weaker model-theoretic notion of algebraicity. We find, for example, that the class of hereditarily ordinal algebraic sets is the same as the class of hereditarily ordinal definable sets; that is, $\mathrm{HOA}=\mathrm{HOD}$. Moreover, we show that every (pointwise) algebraic model of $\mathrm{ZF}$ is actually pointwise definable. Finally, we consider the implicitly constructible universe Imp—an algebraic analogue of the constructible universe—which is obtained by iteratively adding not only the sets that are definable over what has been built so far, but also those that are algebraic (or, equivalently, implicitly definable) over the existing structure. While we know that $\mathrm{Imp}$ can differ from $L$, the subtler properties of this new inner model are just now coming to light. Many questions remain open.