# The implicitly constructible universe

• M. J.~Groszek and J. D. Hamkins, “The implicitly constructible universe,” ArXiv e-prints, 2017. (under review)
@ARTICLE{GroszekHamkins:The-implicitly-constructible-universe,
author = {Marcia J.~Groszek and Joel David Hamkins},
title = {The implicitly constructible universe},
journal = {ArXiv e-prints},
year = 2017,
month = feb,
volume = {},
number = {},
pages = {},
month = {},
note = {under review},
abstract = {},
keywords = {under-review},
source = {},
doi = {},
eprint = {1702.07947},
archivePrefix = {arXiv},
primaryClass = {math.LO},
url = {http://jdh.hamkins.org/the-implicitly-constructible-universe},
}

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