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 $2^\kappa =\kappa ^{++}$ .
Highlights
In [2, 3], Cohen invented the method of forcing as a mean to prove the independence of mathematical propositions from ZFC
Cohen proves that P satisfies the countable chain condition and shows that this condition ensures that the cardinals structure of M[G] is identical to that of M
The first successful transfinite iteration scheme was devised by Solovay and Tennenbaum in [31], who solved a problem concerning a particular type of linear orders of size א1 known as Souslin lines
Summary
In [2, 3], Cohen invented the method of forcing as a mean to prove the independence of mathematical propositions from ZFC (the Zermelo-Fraenkel axioms for set theory). The first successful transfinite iteration scheme was devised by Solovay and Tennenbaum in [31], who solved a problem concerning a particular type of linear orders of size א1 known as Souslin lines They found a natural ccc poset PL to “kill” a given Souslin line L, proved that a (finite-support) iteration of ccc posets is again ccc, and proved that in an iteration of length א2, any Souslin line in the final model must show up in one of the intermediate models, meaning that they can ensure that, in their final model, there are no Souslin lines. In a sequel to this paper [18], we present our iteration scheme for Σ-Prikry notions of forcing, from which we obtain a correct proof of (a strong form of) Sharon’s result: Theorem 1.2 Suppose that ⟨κn ∣ n < ω⟩ is a strictly increasing sequence of Laverindestructible supercompact cardinals.
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