Prikry forcing at κ ⁺ and beyond

E-ISSN： 1943-5886|52|1|44-50

ISSN： 0022-4812

Source： The Journal of Symbolic Logic, Vol.52, Iss.1, 1987-03, pp. : 44-50

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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.