Prikry-forcing and some generalizations (Q2726224)

\textit{K. Prikry} [Diss. Math. 68, 52 p. (1970; Zbl 0212.32404)] devised a simple yet elegant forcing notion for changing the cofinality of a measurable cardinal \(\kappa\) to \(\omega\) without collapsing any cardinals. The generic \(\omega\)-sequence cofinal in \(\kappa\) is called a Prikry sequence. Prikry's method has become an archetype for forcing notions involving large cardinals and is particularly relevant for building models for the failure of the singular cardinal hypothesis SCH which says that \(\lambda^{\text{cf} (\lambda)} = \max \{ \lambda^+ , 2^{\text{cf} (\lambda)} \}\) for singular cardinals \(\lambda\). NEWLINENEWLINENEWLINEThe author provides a detailed analysis of combinatorial properties of (1) Prikry forcing as well as of two of its relatives, namely (2) a forcing devised by \textit{M. Gitik and M. Magidor} [Set theory of the continuum, Math. Sci. Res. Inst. Publ. 26, 243-279 (1992; Zbl 0788.03066)] to simultaneously adjoin many Prikry sequences to a measurable cardinal \(\kappa\) without adding bounded subsets of \(\kappa\) and, thus, to force that GCH (and SCH) first fails at \(\kappa\), and (3) a forcing constructed by \textit{M. Magidor} [Fundam. Math. 99, 61-71 (1978; Zbl 0378.02033)] to make a measurable \(\kappa\) singular of uncountable cofinality without collapsing cardinals. For all three forcing notions, particular emphasis is laid on decomposing the partial order in a direct (or: pure) and a reflexive (or: apure) part and on proving the corresponding Prikry lemma which says that given a sentence \(\phi\) of the forcing language and a condition \(p\) there is a direct (pure) extension \(q\) of \(p\) deciding \(\phi\). NEWLINENEWLINENEWLINEOne of the results presented says intermediate extensions of the Prikry model \(V[g]\) are all of the form \(V[x]\) where \(x\) is a subsequence of the generic Prikry sequence \(g\), thus giving a parametrization of all intermediate extensions by the reals of \(V[g]\). Another result characterizes systems \(\bar g\) of Prikry sequences adjoined by Gitik-Magidor forcing by a combinatorial condition (in such a way that every \(\bar g\) satisfying this condition essentially is a Gitik-Magidor generic system of Prikry sequences). In the same vein, a combinatorial characterization of the sequence generic for Magidor forcing is given. A similar result was proved for Prikry forcing by \textit{A. R. D. Mathias} [J. Aust. Math. Soc. 15, 409-416 (1973; Zbl 0268.02051)].