Sigma-Prikry forcing I: The Axioms

Abstract

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

Negating the Galvin Property

“…Similarly, the same wish extends to singular cardinals, although in this context one will encounter a shortage of parallels of PFA. A potential strategy to overcome this problem might bear on the abstract iteration scheme for singular cardinals introduced in the Σ-Prikry project by Rinot, Sinapova and the third author [PRS21b,PRS21a,PRS21c].…”

Sigma-Prikry forcing II: Iteration Scheme

“…• Σ = κ n | n < ω is an increasing sequence of Laver-indestructible supercompact cardinals; • κ := sup n<ω κ n , µ := κ + and λ := κ ++ ; • 2 κ = κ + and 2 µ = µ + ; • Γ := {α < µ | ω < cf V (α) < κ}. Under these assumptions, Corollaries 4.11 and 6.1 of [PRS19] read as follows, respectively: Fact 5.4. If (P, ℓ, c) is a Σ-Prikry notion of forcing such that 1l P P μ = κ + , then V P |= Refl(<ω, Γ).…”

“…In the introduction to Part I of this series [PRS19], we described the need for iteration schemes and the challenges involved in devising such a scheme, especially at the level of successor of singular cardinals. The main tool available to obtain consistency results at the level of singular cardinals and their successors is the method of forcing with large cardinals, and in particular, Prikry-type forcing.…”

“…Organization of this paper. We assume no familiarity with [PRS19]. In Section 2, we recall the definitions of the Σ-Prikry class, and forking projections.…”

“…See[PRS19, Corollary 5.16] and note that its proof only relies on Clauses (1) and (5) of Definition 2.7.…”