Countable Support Iteration Research Articles

Cohen preservation and independence

We provide a general preservation theorem for preserving selective independent families along countable support iterations. The theorem gives a general framework for a number of results in the literature concerning models in which the independence number i is strictly below c, including iterations of Sacks forcing, Miller partition forcing, h-perfect tree forcings, coding with perfect trees. Moreover, applying the theorem, we show that i=ℵ1 in the Miller Lite model. An important aspect of the preservation theorem is the notion of “Cohen preservation”, which we discuss in detail.

A Selection Principle and Products in Topological Groups

We consider the preservation under products, finite powers, and forcing of a selection-principle-based covering property of T0 topological groups. Though the paper is partly a survey, it contributes some new information: (1) The product of a strictly o-bounded group with an o-bounded group is an o-bounded group—Corollary 1. (2) In the generic extension by a finite support iteration of ℵ1 Hechler reals the product of any o-bounded group with a ground model ℵ0 bounded group is an o-bounded group—Theorem 11. (3) In the generic extension by a countable support iteration of Mathias reals the product of any o-bounded group with a ground model ℵ0 bounded group is an o-bounded group—Theorem 12.

TREE FORCING AND DEFINABLE MAXIMAL INDEPENDENT SETS IN HYPERGRAPHS

AbstractWe 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 $\mathbf {\Delta }^1_2$ maximal independent set. This extends an earlier result by Schrittesser (see [25]). As a main application we get the consistency of $\mathfrak {r} = \mathfrak {u} = \mathfrak {i} = \omega _2$ together with the existence of a $\Delta ^1_2$ ultrafilter, a $\Pi ^1_1$ maximal independent family, and a $\Delta ^1_2$ Hamel basis. This solves open problems of Brendle, Fischer, and Khomskii [5] and the author [23]. We also show in ZFC that $\mathfrak {d} \leq \mathfrak {i}_{cl}$ , addressing another question from [5].

On well-splitting posets

We introduce a class of proper posets which is preserved under countable support iterations, includes $\omega^\omega$-bounding, Cohen, Miller, and Mathias posets associated to filters with the Hurewicz covering properties, and has the property that the ground model reals remain splitting and unbounded in corresponding extensions. Our results may be considered as a possible path towards solving variations of the famous Roitman problem.

Playing with Mathias

We present a series of axioms that describe the combinatorial core of the Mathias model (countable support iteration of ω2 Mathias reals over the Continuum Hypothesis). Our axioms are formulated in terms of games with Borel sets and functions and are strong enough to imply most of the propositions usually proved by means of the iterated Mathias forcing.

The diamond covering property axiom

The Covering Property Axiom, which attempts to capture some of the combinatorics of the Sacks model, the model obtained from by countable support iteration of length of the Sacks forcing, seems to miss a Suslin tree. We add a diamond polish to the axiom to remedy this.

PRESERVATION OF SUSLIN TREES AND SIDE CONDITIONS

AbstractWe show how to force, with finite conditions, the forcing axiom PFA(T), a relativization of PFA to proper forcing notions preserving a given Suslin 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 using a standard countable support iteration and a preservation theorem due to Miyamoto.

Forcing with Sequences of Models of Two Types

We present an approach to forcing with finite sequences of models that uses models of two types. This approach builds on earlier work of Friedman and Mitchell on forcing to add clubs in cardinals larger than ℵ1, with finite conditions. We use the two-type approach to give a new proof of the consistency of the proper forcing axiom. The new proof uses a finite support forcing, as opposed to the countable support iteration in the standard proof. The distinction is important since a proof using finite supports is more amenable to generalizations to cardinals greater than ℵ1.

Understanding preservation theorems: chapter VI of Proper and Improper Forcing, I

We present an exposition of Section VI.1 and most of Section VI.2 from Shelah’s book Proper and Improper Forcing. These sections offer proofs of the preservation under countable support iteration of proper forcing of various properties, including proofs that $${\omega^\omega}$$ -bounding, the Sacks property, the Laver property, and the P-point property are preserved by countable support iteration of proper forcing.

Many countable support iterations of proper forcings preserve Souslin trees

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 iterations in the so-called Case A that does not need a division into forcings that add reals and those who do not.

A Sacks real out of nowhere

AbstractThere is a proper countable support iteration of length ω adding no new reals at finite stages and adding a Sacks real in the limit.

Understanding preservation theorems, II

We present an exposition of much of Sections VI.3 and XVIII.3 from Shelah's book Proper and Improper Forcing. This covers numerous preservation theorems for countable support iterations of proper forcing, including preservation of the property “no new random reals over V ”, the property “reals of the ground model form a non-meager set”, the property “every dense open set contains a dense open set of the ground model”, and preservation theorems related to the weak bounding property, the weak ωω -bounding property, and the property “the set of reals of the ground model has positive outer measure” (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Large continuum, oracles

Abstract Our main theorem is about iterated forcing for making the continuum larger than ℵ2. We present a generalization of [2] which deal with oracles for random, (also for other cases and generalities), by replacing ℵ1,ℵ2 by λ, λ + (starting with λ = λ <λ > ℵ1). Well, we demand absolute c.c.c. So we get, e.g. the continuum is λ + but we can get cov(meagre) = λ and we give some applications. As in non-Cohen oracles [2], it is a “partial” countable support iteration but it is c.c.c.

Applications of the ergodic iteration theorem

I prove several natural preservation theorems for the countable support iteration. This solves a question of Roslanowski regarding the preservation of localization properties and greatly simplifies the proofs in the area (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Cardinal characteristics and projective wellorders

Using countable support iterations of S -proper posets, we show that the existence of a Δ 3 1 definable wellorder of the reals is consistent with each of the following: d < c , b < a = s and b < g .

Decisive creatures and large continuum

AbstractFor f, g ∈ ωω let be the minimal number of uniform g-splitting trees (or: Slaloms) to cover the uniform f-splitting tree, i.e., for every branch v of the f-tree, one of the g-trees contains v. is the dual notion: For every branch v, one of the g-trees guesses v(m) infinitely often.It is consistent that for ℵ1 many pairwise different cardinals κ∊ and suitable pairs (f∊, g∊).For the proof we use creatures with sufficient bigness and halving. We show that the lim-inf creature forcing satisfies fusion and pure decision. We introduce decisiveness and use it to construct a variant of the countable support iteration of such forcings, which still satisfies fusion and pure decision.

New reals: Can live with them, can live without them

We give a self-contained proof of the preservation theorem for proper countable support iterations known as "tools-preservation," "Case A" or "first preservation theorem" in the literature. We do not assume that the forcings add reals.

Preserving preservation

AbstractWe prove that the property “P doesn't make the old reals Lebesgue null” is preserved under countable support iterations of proper forcings, under the additional assumption that the forcings are nep (a generalization of Suslin proper) in an absolute way. We also give some results for general Suslin ccc ideals.

Descriptive set theory and definable forcing

Introduction Definable forcing adding a single real The countable support iterations Other forcings Applications Examples of cardinal invariants The syntax of cardinal invariants Effective descriptive set theory Large cardinals.

On iterating semiproper preorders

AbstractLet T be an ω1-Souslin tree. We show the property of forcing notions; “is {ω1}-semi-proper and preserves T” is preserved by a new kind of revised countable support iteration of arbitrary length. As an application we have a forcing axiom which is compatible with the existence of an ω1 -Souslin tree for preorders as wide as possible.