Magidor and extender Prikry forcing are orientation, not substitutes

Ordinary Prikry forcing uses one normal measure and, as Prikry forcing preserves every cardinal proves, changes the cofinality of its measurable cardinal to $\omega$ without collapsing cardinals. Two related families explain the surrounding landscape but supply none of the later Gitik arguments on this page.

Magidor forcing. With an appropriate coherent sequence of measures and the corresponding Mitchell-order hypothesis, Magidor forcing can change the cofinality of the large cardinal to a prescribed smaller regular cardinal. The word “prescribed” does not mean that every regular target is available from one fixed measurable-cardinal hypothesis: the measure sequence must be long enough for the chosen target. Poveda Ruzafa, Section 7.1, states this parameter and hypothesis explicitly.

Extender-based Prikry forcing. Extender-based variants coordinate many measure projections rather than using the single normal measure of ordinary Prikry forcing. Merimovich's Prikry on Extenders, Revisited is a concrete example: its forcing changes cofinality to $\omega$ without adding bounded subsets while also controlling the power set. “Extender-based” is therefore a family label, not a claim that every such forcing has one common cofinality or preservation profile.

Neither family is a substitute for the construction below. Gitik's target here uses a proper class of strongly compact cardinals, fine complete ultrafilters as in Fine measures, strong compactness and supercompactness, a definable proper-class forcing relation, and a finite-support symmetric submodel. No later item cites this remark as a proof of any of those features.