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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:dobrinen@logic.univie.ac.at, 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:sdf@logic.univie.ac.at, 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

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