Sigma-Prikry forcing I: The Axioms

Author:

Poveda Alejandro,Rinot Assaf,Sinapova Dima

Abstract

AbstractWe introduce a class of notions of forcing which we call $\Sigma $ -Prikry, and show that many of the known Prikry-type notions of forcing that centers around singular cardinals of countable cofinality are $\Sigma $ -Prikry. We show that given a $\Sigma $ -Prikry poset $\mathbb P$ and a name for a non-reflecting stationary set T, there exists a corresponding $\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 $\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 $2^\kappa =\kappa ^{++}$ .

Publisher

Canadian Mathematical Society

Subject

General Mathematics

Cited by 6 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Club stationary reflection and other combinatorial principles at ℵ+2;Annals of Pure and Applied Logic;2025-01

2. Two results on extendible cardinals;Proceedings of the American Mathematical Society;2024-04-29

3. Sigma-Prikry forcing III: Down to ℵ;Advances in Mathematics;2023-12

4. STATIONARY REFLECTION AND THE FAILURE OF THE SCH;The Journal of Symbolic Logic;2023-10-27

5. Negating the Galvin property;Journal of the London Mathematical Society;2023-04-24

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