### A Covering Lemma for HOD of K(ℝ)

Daniel W. Cunningham
Source: Notre Dame J. Formal Logic Volume 51, Number 4 (2010), 427-442.

#### Abstract

Working in ZF+AD alone, we prove that every set of ordinals with cardinality at least Θ can be covered by a set of ordinals in HOD of K(ℝ) of the same cardinality, when there is no inner model with an ℝ-complete measurable cardinal. Here ℝ is the set of reals and Θ is the supremum of the ordinals which are the surjective image of ℝ.

First Page:
Primary Subjects: 03E15
Secondary Subjects: 03E45, 03E60
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.

Permanent link to this document: http://projecteuclid.org/euclid.ndjfl/1285765797
Digital Object Identifier: doi:10.1215/00294527-2010-027
Zentralblatt MATH identifier: 05822355
Mathematical Reviews number (MathSciNet): MR2741835

### References

[1] Barwise, J., Admissible Sets and Structures, Springer-Verlag, Berlin, 1975.
Mathematical Reviews (MathSciNet): MR0424560
Zentralblatt MATH: 0316.02047
[2] Cunningham, D. W., "The real core model and its scales", Annals of Pure and Applied Logic, vol. 72 (1995), pp. 213--89.
Mathematical Reviews (MathSciNet): MR1327117
Zentralblatt MATH: 0828.03025
Digital Object Identifier: doi:10.1016/0168-0072(94)00023-V
[3] Cunningham, D. W., Is there a set of reals not in $K(\R)$?'' Annals of Pure and Applied Logic, vol. 92 (1998), pp. 161--210.
Mathematical Reviews (MathSciNet): MR1625890
Zentralblatt MATH: 0932.03059
Digital Object Identifier: doi:10.1016/S0168-0072(98)00003-7
[4] Cunningham, D. W., "A covering lemma for $L({\R})$", Archive for Mathematical Logic, vol. 41 (2002), pp. 49--54.
Mathematical Reviews (MathSciNet): MR1883109
Zentralblatt MATH: 1022.03031
Digital Object Identifier: doi:10.1007/s001530200003
[5] Cunningham, D. W., "A covering lemma for $K(\R)$", Archive for Mathematical Logic, vol. 46 (2007), pp. 197--221.
Mathematical Reviews (MathSciNet): MR2306176
Zentralblatt MATH: 1110.03044
Digital Object Identifier: doi:10.1007/s00153-007-0040-8
[6] Jech, T., Set Theory, 3d millennium edition, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003.
Mathematical Reviews (MathSciNet): MR1940513
Zentralblatt MATH: 1007.03002
[7] Kanamori, A., The Higher Infinite, Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1994.
Mathematical Reviews (MathSciNet): MR1321144
Zentralblatt MATH: 0813.03034
[8] Kechris, A. S., "Determinacy and the structure of $L({\R})$", pp. 271--83 in Recursion Theory, vol. 42 of Proceedings of Symposia in Pure Mathematics, American Mathematical Society, Providence, 1985.
Mathematical Reviews (MathSciNet): MR791063
Zentralblatt MATH: 0573.03027
[9] Kunen, K., Set Theory, vol. 102 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1980.
Mathematical Reviews (MathSciNet): MR597342
Zentralblatt MATH: 0443.03021
[10] Moschovakis, Y. N., Descriptive Set Theory, vol. 100 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1980.
Mathematical Reviews (MathSciNet): MR561709
Zentralblatt MATH: 0433.03025
[11] Steel, J. R., "Scales in $\mathbf{K}(\mathbb{R})$", pp. 176--208 in Games, Scales, and Suslin Cardinals. The Cabal Seminar. Vol. I, vol. 31 of Lecture Notes in Logic, Association for Symbolic Logic, Chicago, 2008.
Mathematical Reviews (MathSciNet): MR2463612
Zentralblatt MATH: 1167.03032
[12] Woodin, H., handwritten notes.