# The implicitly constructible universe

• M. J. Groszek and J. D. Hamkins, “The implicitly constructible universe,” Journal of Symbolic Logic, vol. 84, iss. 4, p. 1403–1421, 2019.
[Bibtex]
@ARTICLE{GroszekHamkins2019:The-implicitly-constructible-universe,
AUTHOR = {Groszek, Marcia J. and Hamkins, Joel David},
TITLE = {The implicitly constructible universe},
JOURNAL = {Journal of Symbolic Logic},
FJOURNAL = {The Journal of Symbolic Logic},
VOLUME = {84},
YEAR = {2019},
NUMBER = {4},
PAGES = {1403--1421},
ISSN = {0022-4812},
MRCLASS = {03E35 (03E45)},
MRNUMBER = {4045982},
DOI = {10.1017/jsl.2018.57},
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 $(\Imp)^{\Imp} \neq \Imp$.