We investigate two variants of splitting tree forcing, their ideals and regularity properties. We prove connections with other well-known notions, such as Lebesgue measurablility, Baire- and Doughnut-property and the Marczewski field. Moreover, we prove that any \emph{absolute} amoeba forcing for splitting trees necessarily adds a dominating real, providing more support to Spinas' and Hein's conjecture that $\add(\ideal{I}_\spl) \leq \mathfrak{b}$.
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.
We show that compact cardinals and {\rm MM} are sensitive to $λ$-closed forcings for arbitrarily large $λ$. This is done by adding 'regressive' $λ$-Kurepa-trees in either case. We argue that the destruction of regressive Kurepa-trees with {\rm MM} requires the use of Namba forcing.
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 prove that various classical tree forcings -- for instance Sacks forcing, Mathias forcing, Laver forcing, Miller forcing and Silver forcing -- preserve the statement that every real has a sharp and hence analytic determinacy. We then lift this result via methods of inner model theory to obtain level-by-level preservation of projective determinacy (PD). Assuming PD, we further prove that projective generic absoluteness holds and no new equivalence classes classes are added to thin projective transitive relations by these forcings.
We investigate properties of trees of height $ω_1$ and their preservation under subcomplete forcing. We show that subcomplete forcing cannot add a new branch to an $ω_1$-tree. We introduce fragments of subcompleteness which are preserved by subcomplete forcing, and use these in order to show that certain strong forms of rigidity of Suslin trees are preserved by subcomplete forcings. Finally, we explore under what circumstances subcomplete forcing preserves Aronszajn trees of height and width $ω_1$. We show that this is the case if CH fails, and if CH holds, then this is the case iff the bounde…
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.
In the paper we probe the possibilities of creating a Kurepa tree in a generic extension of a model of CH plus no Kurepa trees by an omega_1-preserving forcing notion of size at most omega_1. In the first section we show that in the Levy model obtained by collapsing all cardinals between omega_1 and a strongly inaccessible cardinal by forcing with a countable support Levy collapsing order many omega_1-preserving forcing notions of size at most omega_1 including all omega-proper forcing notions and some proper but not omega-proper forcing notions of size at most omega_1 do not create Kurepa tre…
We show that splitting forcing does not have the weak Sacks property below any condition, answering a question of Laguzzi, Mildenberger and Stuber-Rousselle. We also show how some partition results for splitting trees hold or fail and we determine the value of cardinal invariants after an $ω_2$-length countable support iteration of splitting forcing.
We show that many countable support iterations of proper forcings preserve Souslin trees. We establish sufficient conditions in terms of games and we draw connections to other preservation properties. We present a proof of preservation properties in countable support interations in the so-called Case A that does not need a division into forcings that add reals and those who do not.
Suppose that $T^*$ is an $ω_1$-Aronszajn tree with no stationary antichain. We introduce a forcing axiom PFA($T^*$) for proper forcings which preserve these properties of $T^*$. We prove that PFA($T^*$) implies many of the strong consequences of PFA, such as the failure of very weak club guessing, that all of the cardinal characteristics of the continuum are greater than $ω_1$, and the $P$-ideal dichotomy. On the other hand, PFA($T^*$) implies some of the consequences of diamond principles, such as the existence of Knaster forcings which are not stationarily Knaster.
Generalizing the proof for Sacks forcing, we show that the $h$-perfect tree forcing notions introduced by Goldstern, Judah and Shelah preserve selective independent families even when iterated. As a result we obtain new proofs of the consistency of $\mathfrak{i} = \mathfrak{u} < \mathrm{non} (\mathcal N) = \mathrm{cof} (\mathcal N)$ and $\mathfrak{i} < \mathfrak{u} = \mathrm{non} (\mathcal N) = \mathrm{cof}( \mathcal N)$ as well as some related results.
We present several results that rely on arguments involving the combinatorics of "bushy trees". These include the fact that there are arbitrarily slow-growing diagonally noncomputable (DNC) functions that compute no Kurtz random real, as well as an extension of a result of Kumabe in which we establish that there are DNC functions relative to arbitrary oracles that are of minimal Turing degree. Along the way, we survey some of the existing instances of bushy tree arguments in the literature.
We show that after forcing with a countable support iteration or a finite product of Sacks or splitting forcing over $L$, every analytic hypergraph on a Polish space admits a $\mathbfΔ^1_2$ maximal independent set. As a main application we get the consistency of $\mathfrak{r} = \mathfrak{u} = \mathfrak{i} = ω_2$ together with the existence of a $Δ^1_2$ ultrafilter, a $Π^1_1$ maximal independent family and a $Δ^1_2$ Hamel basis. This solves open problems of Brendle, Fischer and Khomskii and the author. We also show in ZFC that $\mathfrak{d} \leq \mathfrak{i}_{cl}$.
In the first part of the paper, we show that if $ω\le κ< λ$ are cardinals, $κ^{<κ} = κ$, and $λ$ is weakly compact, then in $V[\M(κ,λ)]$ the tree property at $λ= κ^{++V[\M(κ,λ)]}$ is indestructible under all $κ^+$-cc forcing notions which live in $V[\Add(κ,λ)]$, where $\Add(κ,λ)$ is the Cohen forcing for adding $λ$-many subsets of $κ$ and $\M(κ,λ)$ is the standard Mitchell forcing for obtaining the tree property at $λ= (κ^{++})^{V[\M(κ,λ)]}$. This result has direct applications to Prikry-type forcing notions and generalized cardinal invariants. In the second part, we assume that $λ$ is superco…
For $f,g\inω^ω$ let $c^\forall_{f,g}$ be the minimal number of uniform $g$-splitting trees needed to cover the uniform $f$-splitting tree, i.e., for every branch $ν$ of the $f$-tree, one of the $g$-trees contains $ν$. Let $c^\exists_{f,g}$ be the dual notion: For every branch $ν$, one of the $g$-trees guesses $ν(m)$ infinitely often. We show that it is consistent that $c^\exists_{f_ε,g_ε}=c^\forall_{f_ε,g_ε}=κ_ε$ for continuum many pairwise different cardinals $κ_ε$ and suitable pairs $(f_ε,g_ε)$. For the proof we introduce a new mixed-limit creature forcing construction.
We present a systematic study of the method of "norms on possibilities" of building forcing notions with keeping their properties under full control. This technique allows us to answer several open problems, but on our way to get the solutions we develop various ideas interesting per se.These include a new iterable condition for ``not adding Cohen reals'' (which has a flavour of preserving special properties of p-points), new intriguing properties of ultrafilters (weaker than being Ramsey but stronger than p-point) and some new applications of variants of the PP--property.
We prove that there are groups in the constructible universe whose automorphism towers are highly malleable by forcing. This is a consequence of the fact that, under a suitable diamond hypothesis, there are sufficiently many highly rigid non-isomorphic Souslin trees whose isomorphism relation can be precisely controlled by forcing.
We analyze the forcing notion $\mathcal P$ of finite matrices whose rows consists of isomorphic countable elementary submodels of a given structure of the form $H_θ$. We show that forcing with this poset adds a Kurepa tree $T$. Moreover, if $\mathcal P_c$ is a suborder of $\mathcal P$ containing only continuous matrices, then the Kurepa tree $T$ is almost Souslin, i.e. the level set of any antichain in $T$ is not stationary in $ω_1$.
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 …
Assuming $\rm PFA$, we shall use internally club $ω_1$-guessing models as side conditions to show that for every tree $T$ of height $ω_2$ without cofinal branches, there is a proper and $\aleph_2$-preserving forcing notion with finite conditions which specialises $T$. Moreover, the forcing has the $ω_1$-approximation property.
We derive a forcing axiom from the conjunction of square and diamond, and present a few applications, primary among them being the existence of super-Souslin trees. It follows that for every uncountable cardinal $λ$, if $λ^{++}$ is not a Mahlo cardinal in Gödel's constructible universe, then $2^λ= λ^+$ entails the existence of a $λ^+$-complete $λ^{++}$-Souslin tree.
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. …
The purpose of this article is to prove that the forcing axiom for completely proper forcings is inconsistent with the Continuum Hypothesis. This answers a longstanding problem of Shelah. The corresponding completely proper forcing which can be constructed using CH is moreover a tree whose square is special off the diagonal. While such trees had previously been constructed by Jensen and Kunen under the assumption of Jensen's diamond principle, this is the first time such a construction has been carried out using the Continuum Hypothesis.
Let chi be the minimum cardinal of a subset of 2^omega that cannot be made convergent by multiplication with a single Toeplitz matrix. By an application of creature forcing we show that s<chi is consistent. We thus answer a question by Vojtas. We give two kinds of models for the strict inequality. The first is the combination of an aleph_2-iteration of some proper forcing with adding aleph_1 random reals. The second kind of models is got by adding delta random reals to a model of MA_{< kappa} for some delta in [aleph_1,kappa). It was a conjecture of Blass that s=aleph_1<chi=kappa holds in such…
We show that if $cf(2^{\aleph_0})=\aleph_1,$ then any non-trivial $\aleph_1$-closed forcing notion of size $\leq 2^{\aleph_0}$ is forcing equivalent to $Add(\aleph_1, 1),$ the Cohen forcing for adding a new Cohen subset of $ω_1.$ We also produce, relative to the existence of suitable large cardinals, a model of $ZFC$ in which $2^{\aleph_0}=\aleph_2$ and all $\aleph_1$-closed forcing notion of size $\leq 2^{\aleph_0}$ collapse $\aleph_2,$ and hence are forcing equivalent to $Add(\aleph_1, 1).$ These results answer a question of Scott Williams from 1978. We also extend a result of Todorcevic and…
We prove that any suitable generalization of Laver forcing to the space $ κ^κ$, for uncountable regular $κ$, necessarily adds a Cohen $κ$-real. We also study a dichotomy and an ideal naturally related to generalized Laver forcing. Using this dichotomy, we prove the following stronger result: if $ κ^{<κ}=κ$, then every $<κ$-distributive tree forcing on $κ^κ$ adding a dominating $κ$-real which is the image of the generic under a continuous function in the ground model, adds a Cohen $κ$-real. This is a contribution to the study of generalized Baire spaces and answers a question from arXiv:1611.08…
We show that $\mathsf{PFA}$ (Proper Forcing Axiom) implies that adding any number of Cohen subsets of $ω$ will not add an $ω_2$-Aronszajn tree or a weak $ω_1$-Kurepa tree, and moreover no $σ$-centered forcing can add a weak $ω_1$-Kurepa tree (a tree of height and size $ω_1$ with at least $ω_2$ cofinal branches). This partially answers an open problem whether ccc forcings can add $ω_2$-Aronszajn or $ω_1$-Kurepa trees. We actually prove more: We show that a consequence of $\mathsf{PFA}$, namely the guessing model principle, $\mathsf{GMP}$, which is equivalent to the ineffable slender tree proper…
Let l^0 and m^0 be the ideals associated with Laver and Miller forcing, respectively. We show that add (l^0) < cov(l^0) and add (m^0) < cov(m^0) are consistent. We also show that both Laver and Miller forcing collapse the continuum to a cardinal <= h .
In this work, we continue the tradition initiated by Geschke, 2011 of viewing the uncountable Borel chromatic number of analytic graphs as cardinal invariants of the continuum. We show that various uncountable Borel chromatic numbers of closed graphs can be consistently different, as well as consistently equal to the continuum. This is done using arguments that are typical to Axiom A forcing notions.
We show how to force, with finite conditions, the forcing axiom PFA(T), a relativization of PFA to proper forcing notions preserving a given Souslin tree T. The proof uses a Neeman style iteration with generalized side conditions consisting of models of two types, and a preservation theorem for such iterations. The consistency of this axiom was previously known by the standard countable support iteration, using a preservation theorem due to Miyamoto.
The splitting number can be singular. The key method is to construct a forcing poset with finite support matrix iterations of ccc posets introduced by Blass and the second author "Ultrafilters with small generating sets", Israel J. Math., 65, (1989)
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…
We present natural constructions of trees and gaps using a quite general construction scheme. In particular, we solve a natural problem about $(ω_1, ω_1)$-gaps. As it is well known $(ω_1, ω_1)$-gaps can sometimes be filled in $ω_1$-preserving forcing extensions of the set-theoretic universe. There are two natural conditions, dubbed $S$ and $T$ below, that guarantee the existence of such forcing extensions. The condition $T$ is a natural strengthening of the condition $S$ and was motivated by the numerous analogies between $(ω_1,ω_1)$-gaps and certain trees of height $ω_1.$ It turns out that th…
In this paper we analyse some notions of amoeba for tree forcings. In particular we introduce an amoeba-Silver and prove that it satisfies quasi pure decision but not pure decision. Further we define an amoeba-Sacks and prove that it satisfies the Laver property. We also show some application to regularity properties. We finally present a generalized version of amoeba and discuss some interesting associated questions.
It is consistent that there exists a Souslin tree $T$ such that after forcing with it, $T$ becomes an almost Souslin Kurepa tree. This answers a question of Zakrzewski.
In this work we study the uncountable Borel chromatic numbers, defined by Geschke (2011) as cardinal characteristics of the continuum, of low complexity graphs. We show that a strong form of locally countable graphs with compact totally disconnected set of vertices have Borel chromatic number bounded by the continuum of the ground model in the model obtained by adding $\aleph_2$ Laver reals. From this, we answer a question from Geschke and the second author (2022), and another question from Fisher, Friedman and Khomskii (2014) concerning regularity properties of subsets of the real line.
The Steprans forcing notion arises as a quotient of Borel sets modulo the ideal of $σ$-continuity of a certain Borel not $σ$-continuous function. We give a characterization of this forcing in the language of trees and using this characterization we establish such properties of the forcing as fusion and continuous reading of names. Although the latter property is usually implied by the fact that the associated ideal is generated by closed sets, we show it is not the case with Steprans forcing. We also establish a connection between Steprans forcing and Miller forcing thus giving a new descripti…
This dissertation surveys several topics in the general areas of iterated forcing, infinite combinatorics and set theory of the reals. There are two parts. In the first half I consider alternative versions of the Cichoń diagram. First I show that for a wide variety of reduction concepts there is a Cichoń diagram for effective cardinal characteristics relativized to that reduction. As an application I investigate in detail the Cichoń diagram for degrees of constructibility relative to a fixed inner model of ZFC. Then I study generalizations of cardinal characteristics to the space of functions …
We present a method to iterate finitely splitting lim-sup tree forcings along non-wellfounded linear orders. We apply this method to construct a forcing (without using an inaccessible or amalgamation) that makes all definable sets of reals measurable with respect to a certain (non-ccc) ideal.
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 $ω$.
We investigate various strong notions of rigidity for Souslin trees, separating them under Diamond into a hierarchy. Applying our methods to the automorphism tower problem in group theory, we show under Diamond that there is a group whose automorphism tower is highly malleable by forcing.
We give a complete characterization of the sets of cardinals that in a suitable forcing extension can be the Kurepa spectrum, that is, the set of cardinalities of branches of Kurepa trees. This answers a question of the first named author.
For $f,g\inω\ho$ let $\mycfa_{f,g}$ be the minimal number of uniform $g$-splitting trees needed to cover the uniform $f$-splitting tree, i.e. for every branch $ν$ of the $f$-tree, one of the $g$-trees contains $ν$. $\myc_{f,g}$ is the dual notion: For every branch $ν$, one of the $g$-trees guesses $ν(m)$ infinitely often. It is consistent that $\myc_{f_ε,g_ε}=\mycfa_{f_ε,g_ε}=κ_ε$ for $\al1$ many pairwise different cardinals $κ_ε$ and suitable pairs $(f_ε,g_ε)$. For the proof we use creatures with sufficient bigness and halving. We show that the lim-inf creature forcing satisfies fusion and pu…
In this paper we introduce a tree-like forcing notion extending some properties of the random forcing in the context of the generalised Cantor space and study its associated ideal of null sets and notion of measurability. This issue was also addressed by Shelah ([11, Problem 0.5]) and concerns the definition of a forcing which is $κ^kappa$-bounding, $< κ$-closed and $κ^+$-cc, for $κ$ inaccessible.
