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MORE ON BLURRY HOD

Part of: Set theory

Published online by Cambridge University Press:  07 December 2023

GUNTER FUCHS Open the ORCID record for GUNTER FUCHS [Opens in a new window]
GUNTER FUCHS*
Affiliation:
THE COLLEGE OF STATEN ISLAND THE CITY UNIVERSITY OF NEW YORK (CUNY) 2800 VICTORY BOULEVARD STATEN ISLAND NY 10314, USA and THE GRADUATE CENTER THE CITY UNIVERSITY OF NEW YORK (CUNY) 365 5TH AVENUE, NEW YORK NY 10016, USA URL: www.math.csi.cuny.edu/~fuchs
*
E-mail: gunter.fuchs@csi.cuny.edu
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Abstract

I continue the study of the blurry HOD hierarchy. The technically most involved result is that the theory ZFC + “$\aleph _\omega $ is a strong limit cardinal and $\aleph _{\omega +1}$ is the least leap” is equiconsistent with the theory ZFC + “there is a measurable cardinal.”

Keywords

Ordinal definabilityforcinglarge cardinalsinner models

MSC classification

Primary: 03E35: Consistency and independence results 03E40: Other aspects of forcing and Boolean-valued models 03E45: Inner models, including constructibility, ordinal definability, and core models 03E47: Other notions of set-theoretic definability 03E55: Large cardinals
Type
Article
Information
The Journal of Symbolic Logic , First View , pp. 1 - 32
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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References

Bukovský, L., Characterization of generic extensions of models of set theory . Fundamenta Mathematicae , vol. 83 (1973), pp. 35–46.Google Scholar
Dodd, A. and Jensen, R., The core model . Annals of Mathematical Logic , vol. 20 (1981), pp. 43–75.Google Scholar
Dodd, A. and Jensen, R., The covering lemma for $K$ . Annals of Mathematical Logic , vol. 22 (1982), pp. 1–30.Google Scholar
Dodd, A. and Jensen, R., The covering lemma for $L[U]$ . Annals of Mathematical Logic , vol. 22 (1982), pp. 127–135.Google Scholar
Fuchs, G., A characterization of generalized Příkrý sequences . Archive for Mathematical Logic , vol. 44 (2005), no. 8, pp. 935–971.Google Scholar
Fuchs, G., Blurry definability . Mathematics , vol. 10 (2022), no. 3, Article no. 452.Google Scholar
Fuchs, G., On the consistency strength of critical leaps . Archive for Mathematical Logic (2024), pp. 1–12. https://doi.org/10.1007/s00153-024-00951-4.Google Scholar
Fuchs, G., Hamkins, J. D., and Reitz, J., Set-theoretic geology . Annals of Pure and Applied Logic , vol. 166 (2015), no. 4, pp. 464–501.Google Scholar
Gitik, M., Prikry-type forcings , Handbook of Set Theory , vol. 2 (Foreman, M. and Magidor, M., editors), chapter 16, Springer, Berlin, Heidelberg, Dordrecht, New York, 2010, pp. 1351–1447.Google Scholar
Gitik, M., Kanovei, V., and Koepke, P., Intermediate models of Prikry-generic extensions, preprint, 2010. Available at http://www.math.tau.ac.il/˜gitik/.Google Scholar
Hamkins, J. D., Extensions with the approximation and cover properties have no new large cardinals . Fundamenta Mathematicae , vol. 180 (2003), no. 3, pp. 257–277.Google Scholar
Hamkins, J. D. and Leahy, C., Algebraicity and implicit definability in set theory . Notre Dame Journal of Formal Logic , vol. 57 (2016), no. 3, pp. 431–439.Google Scholar
Jensen, R., Definable sets of minimal degree , Mathematical Logic and Foundations of Set Theory (Bar-Hillel, Y., editor), Studies in Logic and the Foundations of Mathematics, vol. 59, Elsevier, Amsterdam, 1970, pp. 122–128.Google Scholar
Kanamori, A., The Higher Infinite , Springer, Dordrecht, Berlin-Heidelberg, 1994.Google Scholar
Kanovei, V. G. and Lyubetsky, V. A., A countable definable set containing no definable elements . Mathematical Notes , vol. 102 (2017), no. 3, pp. 338–349.Google Scholar
Kunen, K., Some applications of iterated ultrapowers in set theory . Annals of Mathematical Logic , vol. 1 (1970), no. 2, pp. 179–227.Google Scholar
Magidor, M., On the singular cardinals problem . I., Israel Journal of Mathematics , vol. 28 (1977), pp. 1–31.Google Scholar
Mathias, A. R. D., On sequences generic in the sense of Prikry . Journal of the Australian Mathematical Society , vol. 15 (1973), pp. 409–414.Google Scholar
Mitchell, W. J., The covering lemma , Handbook of Set Theory , vol. 3 , Springer, Berlin, Heidelberg, Dordrecht, New York, 2010, pp. 1496–1594.Google Scholar
Mitchell, W. J., Beginning Inner Model Theory , Handbook of Set Theory, vol. 3 , Springer, Berlin, Heidelberg, Dordrecht, New York, pp. 1449–1495.Google Scholar
Příkrý, K., Changing measurable into accessible cardinals . Dissertationes Mathematicae (Rozprawy Matematyczne) , vol. 68 (1970), pp. 5–52.Google Scholar
Schimmerling, E., A core model toolbox and guide , Handbook of Set Theory , vol. 3 , Springer, Berlin, Heidelberg, Dordrecht, New York, 2010, pp. 1685–1751.Google Scholar
Solovay, R. M., A model of set-theory in which every set of reals is Lebesgue measurable . Annals of Mathematics. Second Series , vol. 92 (1970), no. 1, pp. 1–56.Google Scholar
Tzouvaras, A., Typicality à la Russell in Set Theory . Notre Dame Journal of Formal Logic , vol. 63 (2022), no. 2, pp. 185–196.Google Scholar