The implicitly constructible universe

  • M. J. Groszek and J. D. Hamkins, “The implicitly constructible universe,” Journal of Symbolic Logic, vol. 84, iss. 4, pp. 1403-1421, 2019.  
    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 = {},

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 $(\Imp)^{\Imp} \neq \Imp$.

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