Adding closed cofinal sequences to large cardinals

Abstract

If κ is measurable, Prikry's forcing adds a sequence of ordinals of order type ω cofinal in κ. This destroys the regularity of κ but κ does remain uncountable. Magidor has a forcing notion generalizing Prikry's which adds a closed cofinal sequence of ordinals through a large cardinal. The cardinal remains uncountable but uts regularity is still destroyed. We obtain a forcing notion which adds a closed cofinal sequence of ordinals (and more complex objects) through a large cardinal κ, of order type κ, and keeps κ regular. In fact κ remains measurable after the forcing.

Our forcing shares certain properties with Prikry's forcing. Closed cofinal sebsequences of generic sequences are generic (under appropriate interpretations). Archetypical generic sequences can be generated by taking the critical points of iterated elementary embeddings.