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Co-stationarity of the ground model

Published online by Cambridge University Press:  12 March 2014

Natasha Dobrinen  and
Sy-David Friedman
Natasha Dobrinen
Affiliation:
Kurt Gödel Research Center For Mathematical Logic, Universität Wien, Währingerstrasse 25, 1090 Wien, Austria.E-mail:[email protected], URL: http://www.logic.univie.ac.at/~dobrinen/
Sy-David Friedman
Affiliation:
Kurt Gödel Research Center For Mathematical Logic, Universität Wien, Währingerstrasse 25, 1090 Wien, Austria.E-mail:[email protected], URL: http://www.logic.univie.ac.at/~sdf/
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Abstract

This paper investigates when it is possible for a partial ordering ℙ to force Pk(Λ)\V to be stationary in Vℙ. It follows from a result of Gitik that whenever ℙ adds a new real, then Pk(Λ)\V is stationary in Vℙ for each regular uncountable cardinal κ in Vℙ and all cardinals λ ≥ κ in Vℙ [4], However, a covering theorem of Magidor implies that when no new ω-sequences are added, large cardinals become necessary [7]. The following is equiconsistent with a proper class of ω1-Erdős cardinals: If ℙ is ℵ1-Cohen forcing, then Pk(Λ)\V is stationary in Vℙ, for all regular κ ≥ ℵ2and all λ ≩ κ. The following is equiconsistent with an ω1-Erdős cardinal: If ℙ is ℵ1-Cohen forcing, then is stationary in Vℙ. The following is equiconsistent with κ measurable cardinals: If ℙ is κ-Cohen forcing, then is stationary in Vℙ.

Keywords

PkΛco-stationarityErdos cardinalmeasurable cardinal
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 71 , Issue 3 , September 2006 , pp. 1029 - 1043
Copyright
Copyright © Association for Symbolic Logic 2006

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References

REFERENCES

[1]Baumgartner, James E., On the size of closed unbounded sets, Annals of Pure and Applied Logic, vol. 54 (1991), pp. 195–227.CrossRefGoogle Scholar
[2]Beller, A., Jensen, R., and Welch, P., Coding the universe, Cambridge University Press, 1982.CrossRefGoogle Scholar
[3]Bukovsky, Lev and Coplakova, Eva, Minimal collapsing extensions of models of ZFC, Annals of Pure and Applied Logic, vol. 46 (1990), pp. 265–298.CrossRefGoogle Scholar
[4]Gitik, Moti, Nonsplitting subsets of , this Journal, vol. 50 (1985), no. 4, pp. 881–894.Google Scholar
[5]Koepke, Peter, Some applications of short cone models, Annals of Pure and Applied Logic, vol. 37 (1988), pp. 179–204.CrossRefGoogle Scholar
[6]Koppelberg, Sabine, Hanbook of Boolean algebra, vol. 1, North-Holland, 1989.Google Scholar
[7]Magidor, Menachem, Representing sets of ordinals as countable unions of sets in the core model, Transactions of American Mathematical Society, vol. 317 (1990), no. 1, pp. 91–126.CrossRefGoogle Scholar
[8]Menas, Telis K., On strong compactness and supercompactness, Annals of Mathematical Logic, vol. 7 (1974/1975), pp. 387–359.CrossRefGoogle Scholar
[9]Mitchell, William, The covering lemma, Handbook of set theory (Foreman, M., Kanamori, A., and Magidor, M., editors), Kluwer Academic Publishers, Dordrecht, to appear.Google Scholar
[10]Namba, Kanji, Independence proof of (ω1, ωα)-distributive law in complete Boolean algebras, Commentarii Mathematici Universitatis Sancti Pauli, vol. 19 (1971), pp. 1–12.Google Scholar
[11]Shelah, Saharon, Independence of strong partition relation for small cardinals, and the free-subset problem, this Journal, vol. 45 (1980), no. 3, pp. 505–509.Google Scholar
[12]Shelah, Saharon, Proper and improper forcing, second ed., Springer-Verlag, 1998.CrossRefGoogle Scholar