Sigma-Prikry forcing I: The Axioms

Joint work with Alejandro Poveda and Dima Sinapova.

Abstract. We introduce a class of notions of forcing which we call $\mathrm{\Sigma }$-Prikry, and show that many of the known Prikry-type notions of forcing that centers around singular cardinals of countable cofinality are $\mathrm{\Sigma }$-Prikry. We show that given a $\mathrm{\Sigma }$-Prikry poset $\mathbb{P}$ and a name for a non-reflecting stationary set $T$, there exists a corresponding $\mathrm{\Sigma }$-Prikry poset that projects to $\mathbb{P}$ and kills the stationarity of $T$. Then, in a sequel to this paper, we develop an iteration scheme for $\mathrm{\Sigma }$-Prikry posets. Putting the two works together, we obtain a proof of the following.

Theorem. If $\kappa$ is the limit of a countable increasing sequence of supercompact cardinals, then there exists a forcing extension in which $\kappa$ remains a strong limit cardinal, every finite collection of stationary subsets of ${\kappa }^{+}$ reflects simultaneously, and $2^{\kappa }={\kappa }^{++}$.

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A. Poveda, A. Rinot and D. Sinapova, Sigma-Prikry forcing I: The axioms, Canad. J. Math., 73(5): 1205-1238, 2021.