Prikry forcing at κ+ and beyond

Abstract

If U is a normal measure on κ then we can add indiscernibles for U either by Prikry forcing [P] or by taking an iterated ultrapower which will add a sequence of indiscernibles for over M. These constructions are equivalent: the set C of indiscernibles for added by the iterated ultrapower is Prikry generic for [Mat]. Prikry forcing has been extended for sequences of measures of length by Magidor [Mag], and his method readily extends to . In this case the measure U is replaced by a sequence of measures and the set C of indiscernibles is replaced by a system of indiscernibles for : is a function such that (κ, β) is a set of indiscernibles for (κ, β) for each . The equivalence between forcing and iterated ultra-powers still holds true for such sequences: there is an interated ultrapower j: V → M (which is defined in detail later in this paper) such that the system of indiscernibles for j() constructed by j is Magidor generic over M.The construction of the system of indiscernibles works equally well for o(κ) ≧ κ+. Radin has defined a variant of Prikry forcing which also works for o(κ) > κ+ ([R]; see also [Mi82] where Radin forcing is applied specifically to sequences of measures, rather than to hypermeasures as in Radin's paper), but Radin's forcing is weaker than Magidor's extension of Prikry forcing in the sense that the system of indiscernibles generated by the interated ultrapower is not Radin generic for j(), but only the set . That is, an indiscernible does not belong to a specific measure, but only to the whole sequence of measures on the cardinal κ.