The implicitly constructible universe

  • M. J.~Groszek and J. D. Hamkins, “The implicitly constructible universe,” ArXiv e-prints, 2017. (manuscript 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 = {manuscript 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 $(\Imp)^{\Imp} \neq \Imp$.