A Characterization of Generalized Příkrý Sequences

A generalization of Příkry'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 Příkry generic sequences reminiscent of Mathias' criterion for Příkry genericity is provided, together with a maximality theorem which states that a generalized Příkry 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. During the research for this paper the author was supported by DFG-Project Je209/1-2.