[bibtex key=”GroszekHamkins2019:The-implicitly-constructible-universe”]

Abstract. We answer several questions posed by Hamkins and Leahy concerning the implicitly constructible universe $\text{Imp}$, which they introduced in their paper, Algebraicity and implicit definability in set theory. Specifically, we show that it is relatively consistent with ZFC that $\text{Imp}\models \mathrm{\neg }\text{CH}$, that $\text{Imp}\ne \text{HOD}$, and that $\text{Imp}\models V\ne \text{Imp}$, or in other words, that $(\text{Imp}{)}^{\text{Imp}}\ne \text{Imp}$.