Abstract
ZF + AD proves that for all nontrivial forcings \(\mathbb{P}\) on a wellorderable set of cardinality less than Θ, \({1_\mathbb{P}}{ \Vdash _\mathbb{P}}\,\neg {\rm{AD}}\). ZF + AD + Θ is regular proves that for all nontrivial forcing \(\mathbb{P}\) which is a surjective image of ℝ, \({1_\mathbb{P}}{ \Vdash _\mathbb{P}}\,\neg {\rm{AD}}\). In particular, ZF + AD + V = L(ℝ) proves that for every nontrivial forcing \( \mathbb{P}\in {L_\Theta }\left(\mathbb{R}\right),{1_\mathbb{P}}{ \Vdash _\mathbb{P}}\,\neg {\rm{AD}}\).
Similar content being viewed by others
References
P. Cohen, The independence of the continuum hypothesis, Proceedings of the National Academy of Sciences of the United States of America 50 (1963), 1143–1148.
S. Jackson, Structural consequences of AD, in Handbook of Set Theory. Vols. 1, 2, 3 Springer, Dordrecht, 2010, pp. 1753–1876.
A. S. Kechris, E. M. Kleinberg, Y. N. Moschovakis and W. H. Woodin, The axiom of determinacy, strong partition properties and nonsingular measures, in Cabal Seminar 77–79 (Proc. Caltech-UCLA Logic Sem., 1977–79), Lecture Notes in Mathematics, Vol. 839, Springer, Berlin-New York, 1981, pp. 75–99.
P. Koellner and W. H. Woodin, Large cardinals from determinacy, in Handbook of Set Theory. Vols. 1, 2, 3, Springer, Dordrecht, 2010, pp. 1951–2119.
P. B. Larson, Extensions of the axiom of determinacy, http://www.users.miamioh.edu/larsonpb/adplusbook_public.pdf.
Y. N. Moschovakis, Descriptive Set Theory, Mathematical Surveys and Monographs, Vol. 155, American Mathematical Society, Providence, RI, 2009.
Author information
Authors and Affiliations
Corresponding author
Additional information
The first author was supported by NSF grant DMS-1703708. The second author was supported by NSF grant DMS-1800323.
Rights and permissions
About this article
Cite this article
Chan, W., Jackson, S. The destruction of the axiom of determinacy by forcings on ℝ when Θ Is regular. Isr. J. Math. 241, 119–138 (2021). https://doi.org/10.1007/s11856-021-2090-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-021-2090-8