We 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 $2^\kappa =\kappa ^{++}$ .

我们引入了一类我们称之为 $\Sigma $ -Prikry的强迫概念 ，并表明许多已知的以可数共尾的奇异基数为中心的普里克里类型的强迫概念是 $\Sigma $ -Prikry。我们证明给定一个 $\Sigma $ -Prikry poset $\mathbb P$ 和一个非反射平稳集T 的名称，存在一个对应的 $\Sigma $ -Prikry poset 投影到 $\mathbb P$ 并杀死T的平稳性。然后，在本文的续篇中，我们为 $\Sigma $ 开发了一个迭代方案 -Prikry 姿势。将这两个工作放在一起，我们得到以下证明。

定理。如果 $\kappa $ 是一个可数递增的超紧基数序列的极限，那么存在一个强迫扩展，其中 $\kappa $ 仍然是一个强极限基数， $\kappa ^+$ 的 每个固定子集的有限集合 同时反映, 和 $2^\kappa =\kappa ^{++}$ 。