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THE IMPLICITLY CONSTRUCTIBLE UNIVERSE

Published online by Cambridge University Press:  10 June 2019

MARCIA J. GROSZEK  and
JOEL DAVID HAMKINS
MARCIA J. GROSZEK
Affiliation:
DEPARTMENT OF MATHEMATICS DARTMOUTH COLLEGE 6188 KEMENY HALL HANOVER, NY03755-3551, USA E-mail: [email protected]
JOEL DAVID HAMKINS
Affiliation:
FACULTY OF PHILOSOPHY UNIVERSITY COLLEGE, OXFORD HIGH STREET, OXFORD OX1 4BH, UK E-mail:[email protected]
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Abstract

We answer several questions posed by Hamkins and Leahy concerning the implicitly constructible universe Imp, which they introduced in [5]. Specifically, we show that it is relatively consistent with ZFC that $$Imp = \neg {\rm{CH}}$$, that $Imp \ne {\rm{HOD}}$, and that $$Imp \models V \ne Imp$$, or in other words, that $\left( {Imp} \right)^{Imp} \ne Imp$.

Keywords

03E3503E45implicitly constructible universe
Type
Articles
Information
The Journal of Symbolic Logic , Volume 84 , Issue 4 , December 2019 , pp. 1403 - 1421
Copyright
Copyright © The Association for Symbolic Logic 2019 

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References

REFERENCES

Abraham, U., A mimimal model for $\neg $CH: Iteration of Jensen’s reals.Transactions of the American Mathematical Society, vol. 281 (1984), pp. 657–674.Google Scholar
Baumgartner, J. and Laver, R., Iterated perfect set forcing. Annals of Pure and Applied Logic, vol. 17 (1979), no. 3, pp. 271–288.Google Scholar
Groszek, M., Applications of iterated perfect set forcing. Annals of Pure and Applied Logic, vol. 39 (1988), no. 1, pp. 19–53.CrossRefGoogle Scholar
Groszek, M. and Jech, T., Generalized iteration of forcing. Transactions of the American Mathematical Society, vol. 324 (1991), pp. 1–26.CrossRefGoogle Scholar
Hamkins, J. D. and Leahy, C., Algebraicity and implicit definability in set theory. Notre Dame Journal of Formal Logic. Advance publication, 20 April 2016. doi: 10.1215/00294527-3542326. http://projecteuclid.org/euclid.ndjfl/1461157794.Google Scholar
Sacks, G. E., Forcing with perfect closed sets, Axiomatic Set Theory (Scott, D., editor), Proceedings of Symposia in Pure Mathematics, vol. 13, American Mathematical Society, Providence, RI, 1971, pp. 331–355.CrossRefGoogle Scholar