We introduce a class of notions of forcing which we call $Σ$-Prikry, and show that many of the known Prikry-type notions of forcing that centers around singular cardinals of countable cofinality are $Σ$-Prikry. We show that given a $Σ$-Prikry poset $\mathbb P$ and a name for a non-reflecting stationary set $T$, there exists a corresponding $Σ$-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 $Σ$-Prikry posets. Putting the two works together, we obtain a proof of the following.
Theorem. If $κ$ is the limit of a countable increasing sequence of supercompact cardinals, then there exists a cofinality-preserving forcing extension in which $κ$ remains a strong limit, every finite collection of stationary subsets of $κ^+$ reflects simultaneously, and $2^κ=κ^{++}$.
In this paper, we answer a question asked in "A minimal Prikry type forcing for singularizing a measurable cardinal" regarding a Mathias criteria for Tree-Prikry forcing. Also we will investigate Prikry forcing using various filters. For completeness and self inclusion reasons, we will give proofs of many known theorems.
Let $D$ be an infinite discrete set of measurable cardinals. It is shown that generalized Prikry forcing to add a countable sequence to each cardinal in $D$ is subcomplete. To do this it is shown that a simplified version of generalized Prikry forcing which adds a point below each cardinal in $D$, called generalized diagonal Prikry forcing, is subcomplete. Moreover, the generalized diagonal Prikry forcing associated to $D$ is subcomplete above $μ$, where $μ$ is any regular cardinal below the first limit point of $D$.
We consider here Easton support iterations of Prikry type forcing notions. New ways of constructing normal ultrafilters in extensions are presented. It turns out that, in contrast with other supports, seemingly unrelated measures or extenders can be involved here.
We study which $κ$-distributive forcing notions of size $κ$ can be embedded into tree Prikry forcing notions with $κ$-complete ultrafilters under various large cardinal assumptions. An alternative formulation -- can the filter of dense open subsets of a $κ$-distributive forcing notion of size $κ$ be extended to a $κ$-complete ultrafilter.
We study the Mathias--Prikry and the Laver type forcings associated with filters and coideals. We isolate a crucial combinatorial property of Mathias reals, and prove that Mathias--Prikry forcings with summable ideals are all mutually bi-embeddable. We show that Mathias forcing associated with the complement of an analytic ideal always adds a dominating real. We also characterize filters for which the associated Mathias--Prikry forcing does not add eventually different reals, and show that they are countably generated provided they are Borel. We give a characterization of $ω$-hitting and $ω$-s…
It is known that the set of possible cofinalities $\mathrm{pcf}(A)$ has good properties if $A$ is a progressive interval of regular cardinals. In this paper, we give an interval of regular cardinals $A$ such that $\mathrm{pcf}(A)$ has no good properties in the presense of a measurable cardinal, or in generic extensions by Prikry-type forcing.
In Part I of this series, we introduced a class of notions of forcing which we call Sigma-Prikry, and showed that many of the known Prikry-type notions of forcing that center around singular cardinals of countable cofinality are Sigma-Prikry. We showed that given a Sigma-Prikry poset P and a P-name for a non-reflecting stationary set T, there exists a corresponding Sigma-Prikry poset that projects to P and kills the stationarity of T. In this paper, we develop a general scheme for iterating Sigma-Prikry posets and, as an application, we blow up the power of a countable limit of Laver-indestruc…
Some models of combinatorial principles have been obtained by collapsing a huge cardinal in the case of the successors of regular cardinals. For example, saturated ideals, Chang's conjecture, polarized partition relations, and transfer principles for chromatic numbers of graphs. In this paper, we study these in the case of the successors of singular cardinals. In particular, we show that Prikry forcing preserves the centeredness of ideals but kills the layeredness. We also study $\binom{μ^{++}}{μ^{+}}\to\binomκ{μ^{+}}_μ$ and $\mathrm{Tr}_{\mathrm{Chr}}(μ^{+++},μ^{+})$ in the extension by Prikr…
We prove the consistency of the failure of the singular cardinals hypothesis at $\aleph_ω$ together with the reflection of all stationary subsets of $\aleph_{ω+1}$. This shows that two classic results of Magidor (from 1977 and 1982) can hold simultaneously.
We continue Gitik, Kanovei and Koepke's work and study sets in generic extensions by the Magidor forcing and by the Prikry forcing with non-normal ultrafilters.
We study the nonstationary-support iteration of Prikry forcings below a measurable cardinal κ, characterizing all the normal measures it carries in the generic extension. We then analyze the restriction of ultrapower embeddings, taken with such a normal measure in the generic extension, to the ground model. We prove that every such restriction is an iterated ultrapwer of the ground model, and provide a sufficient condition for its definability there. This is done without core-model theoretic arguments: the assumption that GCH holds in the ground model up to κsuffices.
I isolate a combinatorial property of a poset $\mathbb{P}$ that I call the strong Prikry property, which implies the existence of an ultrafilter on the complete Boolean algebra $\mathbb{B}$ of $\mathbb{P}$ such that one inclusion of the Boolean ultrapower version of the so-called \Bukovsky-Dehornoy phenomenon holds with respect to $\mathbb{B}$ and $U$. I show that in all cases that were previously studied, and for which it was shown that they come with a canonical iterated ultrapower construction whose limit can be described as a single Boolean ultrapower, the posets in question satisfy this p…
在本文中，我们介绍了几类Prikry型强制概念，其中两个是用来产生最小的一般扩张，第三个是应用在$α$ -递归理论，以产生最小的覆盖。第一个强制作为预热，在可测量基数（在$V$中）上产生最小通用扩展，第二个强制在$ω$ -可测量基数的极限$\langleγ_n\colon n<ω\rangle$，使得每个$γ_n$（$n>0$）携带$γ_{n-1}$ -许多正常测量。通过一个$V_γ$ -度的概念（见定义\ref{def:vgammadegree}），我们将最小类属扩张的第二个Prikry型构造转化为$α$ -递归理论中的最小度构造.更明确地说，\begin{theorem*} Suppose $\langleγ_n\colon n<ω\rangle$ is a strictly increasing sequence of measurable cardinals such that for each $n>0$, $γ_n$ carries at least $γ_{n-1}$-many normal measures. Let $γ=\sup\{γ_n\colon n<ω\}$. %Then for each $n$, $γ$ is $Σ_n$-admissible. Then there is an $A\subsetγ$ such that \begin{itemize}…
We continue the work done by the authors and before that by the second author, Kanovei and koepke. We prove that for every set of ordinals $A$ in a Magidor-Radin generic extension using a coherent sequence such that $o^{\vec{U}}(κ)<κ^+$, there is $C'\subseteq C_G$, such that $V[A]=V[C']$. Also we prove that the supremum of a fresh set in a Prikry, tree Prikry, Magidor, Radin-Magidor and Radin forcing, changes cofinality to $ω$.
Consider $(κ^{+++},κ^{++}) \twoheadrightarrow (κ^+,κ)$ where $κ$ is an uncountable regular cardinal. By a result of Shelah's we have $\operatorname{cof}(X \cap κ^{++}) = κ$ for almost all $X \subset κ^{+++}$ witnessing this. Here we consider the question if there could be a similar result for $X \cap κ^+$. We use this discussion to give an interesting example of a pseudo Prikry forcing answering a question of Sinapova.
Dobrinen, Hathaway and Prikry studied a forcing $\mathbb{P}_κ$ consisting of perfect trees of height $λ$ and width $κ$ where $κ$ is a singular $ω$-strong limit of cofinality $λ$. They showed that if $κ$ is singular of countable cofinality, then $\mathbb{P}_κ$ is minimal for $ω$-sequences assuming that $κ$ is a supremum of a sequence of measurable cardinals. We obtain this result without the measurability assumption. Prikry proved that $\mathbb{P}_κ$ is $(ω,ν)$-distributive for all $ν<κ$ given a singular $ω$-strong limit cardinal $κ$ of countable cofinality, and Dobrinen et al$.$ asked whether …
We investigate forcing properties of perfect tree forcings defined by Prikry to answer a question of Solovay in the late 1960's regarding first failures of distributivity. Given a strictly increasing sequence of regular cardinals $\langle κ_n: n< ω\rangle$, Prikry defined the forcing $\mathbb{P}$ all perfect subtrees of $\prod_{n<ω}κ_n$, and proved that for $κ=\sup_{n<ω}κ_n$, assuming the necessary cardinal arithmetic, the Boolean completion $\mathbb{B}$ of $\mathbb{P}$ is $(ω,μ)$-distributive for all $μ<κ$ but $(ω,κ,δ)$-distributivity fails for all $δ<κ$, implying failure of the $(ω,κ)$-d.l. …
An $\aleph_1$-Souslin tree is a complicated combinatorial object whose existence cannot be decided on the grounds of ZFC alone. But 15 years after Tennenbaum and independently Jech devised notions of forcing for introducing such a tree, Shelah proved that already the simplest forcing notion --- Cohen forcing --- adds an $\aleph_1$-Souslin tree. In this paper, we identify a rather large class of notions of forcing that, assuming a GCH-type assumption, add a $λ^+$-Souslin tree. This class includes Prikry, Magidor and Radin forcing.
We show that while the length $ω$ iterated ultrapower by a normal ultrafilter is a Boolean ultrapower by the Boolean algebra of Prikry forcing, it is consistent that no iteration of length greater than $ω$ (of the same ultrafilter and its images) is a Boolean ultrapower. For longer iterations, where different ultrafilters are used, this is possible, though, and we give Magidor forcing and a generalization of Prikry forcing as examples. We refer to the discovery that the intersection of the finite iterates of the universe by a normal measure is the same as the generic extension of the direct li…
Given a cardinal $λ$, category forcing axioms for $λ$-suitable classes $Γ$ are strong forcing axioms which completely decide the theory of the Chang model $\mathcal C_λ$, modulo generic extensions via forcing notions from $Γ$. $\mathsf{MM}^{+++}$ was the first category forcing axiom to be isolated (by the second author). In this paper we present, without proofs, a general theory of category forcings, and prove the existence of $\aleph_1$-many pairwise incompatible category forcing axioms for $ω_1$-suitable classes.
We prove that the forcing axiom $MA^{1.5}_{\aleph_2}(\mbox{stratified})$ implies $\Box_{ω_1, ω_1}$. Using this implication, we show that the forcing axiom $MM_{\aleph_2}(\aleph_2\mbox{-c.c.})$ is inconsistent. We also derive weak Chang's Conjecture from $MA^{1.5}_{\aleph_2}(\mbox{stratified})$ and use this second implication to give another proof of the inconsistency of $MM_{\aleph_2}(\aleph_2\mbox{-c.c.})$.
We show that the forcing axiom for countably compact, $ω_2$-Knaster, well-met posets is inconsistent. This is supplemental to an inconsistency result of Shelah and sets a new limit to the generalization of Martin's Axiom to the stage of $ω_2$.
Methods of Higher Forcing Axioms was a small workshop in Norwich, taking place between 10--12 of September, 2019. The goal was to encourage future collaborations, and create more focused threads of research on the topic of higher forcing axioms. This is an improved version of the notes taken during the meeting by Asaf Karagila.
We introduce bounded category forcing axioms for well-behaved classes $Γ$. These are strong forms of bounded forcing axioms which completely decide the theory of some initial segment of the universe $H_{λ_Γ^+}$ modulo forcing in $Γ$, for some cardinal $λ_Γ$ naturally associated to $Γ$. These axioms naturally extend projective absoluteness for arbitrary set-forcing--in this situation $λ_Γ=ω$--to classes $Γ$ with $λ_Γ>ω$. Unlike projective absoluteness, these higher bounded category forcing axioms do not follow from large cardinal axioms, but can be forced under mild large cardinal assumptions o…
We note that some form of the condition "$p_1, p_2$ have a $\leq_{\mathbb{Q}}$-lub in $\mathbb{Q}$" is necessary in some forcing axiom for $λ$-complete $μ^+$-c.c. forcing notions. We also show some versions are really stronger than others.
We study principles of the form: if a name $σ$ is forced to have a certain property $φ$, then there is a ground model filter $g$ such that $σ^g$ satisfies $φ$. We prove a general correspondence connecting these name principles to forcing axioms. Special cases of the main theorem are: Any forcing axiom can be expressed as a name principle. For instance, $\mathsf{PFA}$ is equivalent to a principle for rank $1$ names (equivalently, nice names) for subsets of $ω_1$, and a principle for rank $2$ names for sets of reals. Moreover, $λ$-bounded forcing axioms are equivalent to name principles. Bagaria…
A central theme in set theory is to find universes with extreme, well-understood behaviour. The case we are interested in is assuming GCH and has a strong forcing axiom of higher order than usual. Instead of "for every suitable forcing notion for~$λ$" we shall say "for every such family of forcing notions, depending on stationary $S\subseteq λ$, for some such stationary set we have\dots". Such notions of forcing are important for Abelian group theory, but this application is delayed for a sequel.
