Abstract
We show first that it is consistent that κ is a measurable cardinal where the GCH fails, while there is a lightface definable wellorder of H(κ +). Then with further forcing we show that it is consistent that GCH fails at ℵ ω , ℵ ω strong limit, while there is a lightface definable wellorder of H(ℵ ω+1) (“definable failure” of the singular cardinal hypothesis at ℵ ω ). The large cardinal hypothesis used is the existence of a κ ++-strong cardinal, where κ is κ ++-strong if there is an embedding j: V → M with critical point κ such that H(κ ++) ⊆ M. By work of M. Gitik and W. J. Mitchell [12], [20], our large cardinal assumption is almost optimal. The techniques of proof include the “tuning-fork” method of [10] and [3], a generalisation to large cardinals of the stationary-coding of [4] and a new “definable-collapse” coding based on mutual stationarity. The fine structure of the canonical inner model L[E] for a κ ++-strong cardinal is used throughout.
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References
D. Asperó and S. D. Friedman, Large cardinals and the locally defined well-orders of the universe, Annals of Pure and Applied Logic 157 (2009), 1–15.
J. Cummings, Iterated forcing and elementary embeddings, in Handbook of Set Theory (M. Foreman and A. Kanamori, eds.), Vol. 2, Springer, Berlin, 2010.
N. Dobrinen and S. D. Friedman, The consistency strength of the tree property at the double successor of a measurable, Fundamenta Mathematicae 208 (2010), 123–153.
V. Fischer and S. D. Friedman, Cardinal characteristics and projective wellorders, Annals of Pure and Applied Logic 161 (2010), 916–922.
V. Fischer, S. D. Friedman and L. Zdomskyy, Projective wellorders and MAD families with large continuum, Annals of Pure and Applied Logic 162 (2011), 853–862
M. Foreman and W. H. Woodin, GCH can fail everywhere, Annals of Mathematics 133 (1991), 1–35.
S. D. Friedman, Lecture notes on definable wellorders, see http://www.logic.univie.ac.at/~sdf/papers/.
S. D. Friedman and R. Honzik, Easton’s theorem and large cardinals, Annals of Pure and Applied Logic 154 (2008), 191–208.
S. D. Friedman and M. Magidor, The number of normal measures, The Journal of Symbolic Logic 74 (2009), 1069–1080.
S. D. Friedman and K. Thompson, Perfect trees and elementary embeddings, The Journal of Symbolic Logic 73 (2008), 906–918.
S. D. Friedman and L. Zdomskyy, Measurable cardinals and the cofinality of the symmetric group, Fundamenta Mathematicae 207 (2010), 101–122.
M. Gitik, The negation of singular cardinal hypothesis from o(κ) = κ ++, Annals of Pure and Applied Logic 43 (1989), 209–234.
M. Gitik, Prikry-type forcings, in Handbook of Set Theory (M. Foreman and A. Kanamori, eds.), Vol. 2, Springer, Berlin, 2010.
L. Harrington, Long projective wellorderings, Annals of Mathematical Logic 12 (1977), 1–24.
R. Honzik, Global singularization and the failure of SCH, Annals of Pure and Applied Logic 161 (2010), 895–915.
T. Jech, Set theory, Springer, Berlin, 2003.
A. Kanamori, Perfect-set forcing for uncountable cardinals, Annals of Mathematical Logic 19 (1980), 97–114.
A. Kanamori, The Higher Infinite, Springer, Berlin, 2003.
M. Magidor, On the singular cardinals problem I, Israel Journal of Mathematics 28 (1977), 1–31.
W. J. Mitchell, The core model for sequences of measures. I, Mathematical Proceedings of the Cambridge Philosophical Society 95 (1984), 229–260.
W. J. Mitchell and J. R. Steel, Fine structure and iteration trees, Lecture Notes in Logic, Vol. 3, Springer, Berlin, 1994.
S. Shelah, Proper and Improper Forcing, Springer, Berlin, 1998.
J. R. Steel and B. Loewe, An introduction to core model theory, Sets and Proofs, London Mathematical Society Lecture Notes, Vol. 258, Cambridge University Press, 1999.
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The first author was supported by FWF Project P19898-N18.
The second author was supported by postdoctoral grant of the Grant Agency of the Czech Republic 201/09/P115.
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Friedman, SD., Honzik, R. A definable failure of the singular cardinal hypothesis. Isr. J. Math. 192, 719–762 (2012). https://doi.org/10.1007/s11856-012-0044-x
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DOI: https://doi.org/10.1007/s11856-012-0044-x