Abstract
A generalization of Příkrý's forcing is analyzed which adjoins to a model of ZFC a set of order type at most ω below each member of a discrete set of measurable cardinals. A characterization of generalized Příkrý generic sequences reminiscent of Mathias' criterion for Příkrý genericity is provided, together with a maximality theorem which states that a generalized Příkrý sequence almost contains every other one lying in the same extension.
This forcing can be used to falsify the covering lemma for a higher core model if there is an inner model with infinitely many measurable cardinals – changing neither cardinalities nor cofinalities. Another application is an alternative proof of a theorem of Mitchell stating that if the core model contains a regular limit θ of measurable cardinals, then there is a model in which every set of measurable cardinals of K bounded in θ has an indiscernible sequence but there is no such sequence for the entire set of measurables of K below θ.
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During the research for this paper the author was supported by DFG-Project Je209/1-2.
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Fuchs, G. A Characterization of Generalized Příkrý Sequences. Arch. Math. Logic 44, 935–971 (2005). https://doi.org/10.1007/s00153-005-0313-z
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DOI: https://doi.org/10.1007/s00153-005-0313-z