Sigma-prikry forcing I: The axioms

We introduce a class of notions of forcing which we call Σ-Prikry, and show that many of the known Prikry-type notions of forcing that centers around singular cardinals of countable cofinality are Σ-Prikry. We show that given a Σ-Prikry poset and a name for a non-reflecting stationary set T, there exists a corresponding Σ-Prikry poset that projects to and kills the stationarity of T. Then, in a sequel to this paper, we develop an iteration scheme for Σ-Prikry posets. Putting the two works together, we obtain a proof of the following. Theorem. If k is the limit of a countable increasing sequence of supercompact cardinals, then there exists a forcing extension in which remains a strong limit cardinal, every finite collection of stationary subsets of K+ reflects simultaneously, and 2k = k++.