Proper Forcing Axiom Research Articles

Indestructibility of some compactness principles over models of [formula omitted

We show that PFA (Proper Forcing Axiom) implies that adding any number of Cohen subsets of ω will not add an ω2-Aronszajn tree or a weak ω1-Kurepa tree, and moreover no σ-centered forcing can add a weak ω1-Kurepa tree (a tree of height and size ω1 with at least ω2 cofinal branches). This partially answers an open problem whether ccc forcings can add ω2-Aronszajn trees or ω1-Kurepa trees (with ¬□ω1 in the latter case).We actually prove more: We show that a consequence of PFA, namely the guessing model principle, GMP, which is equivalent to the ineffable slender tree property, ISP, is preserved by adding any number of Cohen subsets of ω. And moreover, GMP implies that no σ-centered forcing can add a weak ω1-Kurepa tree (see Section 2.1 for definitions).For more generality, we study variations of the principle GMP at higher cardinals and the indestructibility consequences they entail, and as applications we answer a question of Mohammadpour about guessing models at weakly but not strongly inaccessible cardinals and show that there is a model in which there are no weak ℵω+1-Kurepa trees and no ℵω+2-Aronszajn trees.

Locally compact, ω1-compact spaces

An ω1-compact space is a space in which every closed discrete subspace is countable. We give various general conditions under which a locally compact, ω1-compact space is σ-countably compact, i.e., the union of countably many countably compact spaces. These conditions involve very elementary properties.Many results shown here are independent of the usual (ZFC) axioms of set theory, and the consistency of some may involve large cardinals. For example, it is independent of the ZFC axioms whether every locally compact, ω1-compact space of cardinality ℵ1 is σ-countably compact. Whether ℵ1 can be replaced with ℵ2 is a difficult unsolved problem. Modulo large cardinals, it is also ZFC-independent whether every hereditarily normal, or every monotonically normal, locally compact, ω1-compact space is σ-countably compact.As a result, it is also ZFC-independent whether there is a locally compact, ω1-compact Dowker space of cardinality ℵ1, or one that does not contain both an uncountable closed discrete subspace and a copy of the ordinal space ω1.Set theoretic tools used for the consistency results include the existence of a Souslin tree, the Proper Forcing Axiom (PFA), and models generically referred to as “MM(S)[S]”. Most of the work is done by the P-Ideal Dichotomy (PID) axiom, which holds in the latter two cases, and which requires no large cardinal axioms when directly applied to topological spaces of cardinality ℵ1, as it is in several theorems.

Non-vanishing higher derived limits

In the study of strong homology Mardešić and Prasolov isolated a certain inverse system of abelian groups [Formula: see text] indexed by elements of [Formula: see text]. They showed that if strong homology is additive on a class of spaces containing closed subsets of Euclidean spaces then the higher derived limits [Formula: see text] must vanish, for [Formula: see text]. They also proved that under the Continuum Hypothesis [Formula: see text]. The question whether [Formula: see text] vanishes, for all [Formula: see text], has attracted considerable interest from set theorists. Dow, Simon and Vaughan showed that under the Proper Forcing Axiom (PFA) [Formula: see text]. Bergfalk showed that it is consistent that [Formula: see text] does not vanish. Later Bergfalk and Lambie-Hanson showed that, modulo a weakly compact cardinal, it is relatively consistent with ZFC that [Formula: see text], for all [Formula: see text]. The large cardinal assumption was recently removed by Bergfalk, Hrušak and Lambie-Hanson. We complete the picture by showing that, for any [Formula: see text], it is relatively consistent with ZFC that [Formula: see text].

On the property (C) of Corson and other sequential properties of Banach spaces

A well-known result of R. Pol states that a Banach space X has property (C) of Corson if and only if every point in the weak*-closure of any convex set C⊆BX⁎ is actually in the weak*-closure of a countable subset of C. Nevertheless, it is an open problem whether this is in turn equivalent to the countable tightness of BX⁎ with respect to the weak*-topology. Frankiewicz, Plebanek and Ryll-Nardzewski provided an affirmative answer under MA+¬CH for the class of C(K)-spaces. In this article we provide a partial extension of this latter result by showing that under the Proper Forcing Axiom (PFA) the following conditions are equivalent for an arbitrary Banach space X:(1)X has property E′;(2)X has weak*-sequential dual ball;(3)X has property (C) of Corson;(4)(BX⁎,w⁎) has countable tightness. This provides a partial extension of a former result of Arhangel'skii. In addition, we show that every Banach space with property E′ has weak*-convex block compact dual ball.

Uncountable almost irredundant sets in nonseparable C*-algebras

In this article, we consider the notion of almost irredundant sets: A subset X of a C*-algebra A is called almost irredundant if and only if for every a∈X, the element a does not belong to the norm-closure of{∑i=1nλi∏j=1niai,j:whereai,j∈X∖{a}and∑|λi|≤1}. Since every almost irredundant set is in particular a discrete set, it follows that the density of A is an upper bound for the size of almost irredundant sets. We prove that under the Proper Forcing Axiom (PFA), there is an uncountable almost irredundant set in every C*-algebra with an uncountable increasing sequence of ideals. In particular, assuming PFA, every nonseparable scattered C*-algebra admits an uncountable almost irredundant set.

Distributive proper forcing axiom and a left-right dichotomy of Cichoń’s diagram

In this paper, we study distributive proper forcing axiom (DPFA) and prove its consistency with a dichotomy of the Cichon’s diagram, relative to certain large cardinal assumption. Namely, we evaluate the cardinal invariants in Cichon’s diagram with the first two uncountable cardinals in the way that the left-hand side has the least possible cardinality while the right-hand side has the largest possible value, and preserve the evaluation along the way of forcing DPFA.

PFA and guessing models

This paper explores the consistency strength of The Proper Forcing Axiom (PFA) and the theory (T) which involves a variation of the Viale–Weiβ guessing hull principle. We show that (T) is consistent relative to a supercompact cardinal. The main result of the paper is Theorem 0.2, which implies that the theory “ADR + ⊝ is regular” is consistent relative to (T) and to PFA. This improves significantly the previous known best lower-bound for consistency strength for (T) and PFA, which is roughly “ADR + DC”.

Prevalence of Generic Laver Diamond

Viale \cite{Viale_GuessingModel} introduced the notion of Generic Laver Diamond at $\kappa$---which we denote $\Diamond_{\text{Lav}}(\kappa)$---asserting the existence of a single function from $\kappa \to H_\kappa$ that behaves much like a supercompact Laver function, except with generic elementary embeddings rather than internal embeddings. Viale proved that the Proper Forcing Axiom (PFA) implies $\Diamond_{\text{Lav}}(\omega_2)$. We strengthen his theorem by weakening the hypothesis to a statement strictly weaker than PFA. We also show that the principle $\Diamond_{\text{Lav}}(\kappa)$ provides a uniform, simple construction of 2-cardinal diamonds, and prove that $\Diamond_{\text{Lav}}(\kappa)$ is quite prevalent in models of set theory; in particular: 1) $L$ satisfies $\Diamond_{\text{Lav}}^+(\kappa)$ whenever $\kappa$ is a successor cardinal, or when the appropriate version of Chang's Conjecture fails. 2) For any successor cardinal $\kappa$, there is a $\kappa$-directed closed class forcing---namely, the forcing from Friedman-Holy \cite{MR2860182}---that forces $\Diamond_{\text{Lav}}(\kappa)$.

A quasi-lower bound on the consistency strength of PFA

A long-standing open question is whether supercompactness provides a lower bound on the consistency strength of the Proper Forcing Axiom (PFA). In this article we establish a quasi lower bound by showing that there is a model with a proper class of subcompact cardinals such that PFA (indeed the weaker statement that PFA holds for (2א0)+-linked forcings) fails in all of its proper forcing extensions. Neeman obtained such a result assuming the existence of “fine structural” models containing very large cardinals, however the existence of such models remains open. We show that Neeman’s arguments go through for a similar notion of “L-like” model and establish the existence of Llike models containing very large cardinals. The main technical result needed is the compatibility of Local Club Condensation with Acceptability in the presence of very large cardinals, a result which constitutes further progress in the outer model programme. The core model programme (initiated by Jensen, see Steel’s [16] for a survey) has had considerable success in establishing lower bounds on the consistency strength of set-theoretic statements, up to the level of Woodin cardinals. But the consistency strength of the Proper Forcing Axiom (PFA) is conjectured to be that of a supercompact cardinal, for which no core model theory is currently available. It is therefore worthwhile to consider quasi lower bounds on the consistency strength of PFA and the main result of this paper is that a proper class of subcompact cardinals serves as such a quasi lower bound: Theorem 1. Assuming the consistency of a proper class of subcompact cardinals, it is consistent that there is a proper class of subcompact cardinals, but PFA (even restricted to posets which are (2א0)+-linked) holds in no proper extension of the universe. What exactly is meant by a quasi lower bound? The necessary ingredients are • the desired set-theoretic principle φ for which we want to obtain a quasi-lower bound result 2000 Mathematics Subject Classification. 03E35, 03E55, 03E57.

Distributive proper forcing axiom and cardinal invariants

In this paper, we study the forcing axiom for the class of proper forcing notions which do not add ? sequence of ordinals. We study the relationship between this forcing axiom and many cardinal invariants. We use typical iterated forcing with large cardinals and analyse certain property being preserved in this process. Lastly, we apply the results to distinguish several forcing axioms.

Operations, climbability and the proper forcing axiom

In this paper we show that the Proper Forcing Axiom (PFA) is preserved under forcing over any poset P with the following property: In the generalized Banach–Mazur game over P of length (ω1+1), Player II has a winning strategy which depends only on the current position and the ordinal indicating the number of moves made so far. By the current position we mean: The move just made by Player I for a successor stage, or the infimum of all the moves made so far for a limit stage. As a consequence of this theorem, we introduce a weak form of the square principle and show that it is consistent with PFA.

PFA and Ideals on ω2 Whose Associated Forcings Are Proper

Given an ideal I, let PI denote the forcing with I-positive sets. We consider models of forcing axioms MA(Γ) which also have a normal ideal I with completeness ω2 such that PI∈Γ. Using a bit more than a superhuge cardinal, we produce a model of PFA (proper forcing axiom) which has many ideals on ω2 whose associated forcings are proper; a similar phenomenon is also observed in the standard model of MA+ω1(σ-closed) obtained from a supercompact cardinal. Our model of PFA also exhibits weaker versions of ideal properties, which were shown by Foreman and Magidor to be inconsistent with PFA. Along the way, we also show (1) the diagonal reflection principle for internally club sets (DRP(ICω1)) introduced by the author in earlier work is equivalent to a natural weakening of “there is an ideal I such that PI is proper”; and (2) for many natural classes Γ of posets, MA+ω1(Γ) is equivalent to an apparently stronger version which we call MA+Diag(Γ).

Martin’s Axiom and separated mad families

Two families \(\mathcal{A}, \mathcal{B}\) of subsets of ω are said to be separated if there is a subset of ω which mod finite contains every member of \(\mathcal{A}\) and is almost disjoint from every member of \(\mathcal{B}\). If \(\mathcal{A}\) and \(\mathcal{B}\) are countable disjoint subsets of an almost disjoint family, then they are separated. Luzin gaps are well-known examples of ω1-sized subfamilies of an almost disjoint family which can not be separated. An almost disjoint family will be said to be ω1-separated if any disjoint pair of ≤ω1-sized subsets are separated. It is known that the proper forcing axiom (PFA) implies that no maximal almost disjoint family is ≤ω1-separated. We prove that this does not follow from Martin’s Axiom.

Scott's problem for Proper Scott sets

AbstractSome 40 years ago, Dana Scott proved that every countable Scott set is the standard system of a model of PA. Two decades later, Knight and Nadel extended his result to Scott sets of size ω1. Here, I show that assuming the Proper Forcing Axiom (PFA), every A-proper Scott set is the standard system of a model of PA. I define that a Scott set is proper if the quotient Boolean algebra /Fin is a proper partial order and A-proper if is additionally arithmetically closed. I also investigate the question of the existence of proper Scott sets.

PFA implies ADL(ℝ)

In this paper we shall proveTheorem 0.1. Suppose there is a singular strong limit cardinal κ such that □κ fails; then AD holds in L(R).See [10] for a discussion of the background to this problem. We suspect that more work will produce a proof of the theorem with its hypothesis that κ is a strong limit weakened to ∀α < κ (αω < κ), and significantly more work will enable one to drop the hypothesis that K is a strong limit entirely. At present, we do not see how to carry out even the less ambitious project.Todorcevic [23] has shown that if the Proper Forcing Axiom (PFA) holds, then □κ fails for all uncountable cardinals κ. Thus we get immediately:It has been known since the early 90's that PFA implies PD, that PFA plus the existence of a strongly inaccessible cardinal implies ADL(ℝ) and that PFA plus a measurable yields an inner model of ADℝ containing all reals and ordinals. As we do here, these arguments made use of Tororcevic's work, so that logical strength is ultimately coming from a failure of covering for some appropriate core models.In late 2000, A. S. Zoble and the author showed that (certain consequences of) Todorcevic's Strong Reflection Principle (SRP) imply ADL(ℝ). (See [22].) Since Martin's Maximum implies SRP, this gave the first derivation of ADL(ℝ) from an “unaugmented” forcing axiom.

Completely Additive Liftings

The purpose of this communication is to survey a theory of liftings, as developed in author's thesis ([8]). The first result in this area was Shelah's construction of a model of set theory in which every automorphism ofP(ℕ)/ Fin, where Fin is the ideal of finite sets, istrivial, or inother words, it is induced by a function mapping integers into integers ([33]). (It is a classical result of W. Rudin [31] that under the Continuum Hypothesis there are automorphisms other than trivial ones.) Soon afterwards, Velickovic ([47]), was able to extract from Shelah's argument the fact that every automorphism ofP(ℕ)/ Fin with a Baire-measurable lifting has to be trivial. This, for instance, implies that in Solovay's model ([36]) all automorphisms are trivial. Later on, an axiomatic approach was adopted and Shelah's conclusion was drawn first from the Proper Forcing Axiom (PFA) ([34]) and then from the milder Open Coloring Axiom (OCA) and Martin's Axiom (MA) ([48], see §5 for definitions). Both shifts from the quotientP(ℕ)/ Fin to quotients over more general idealsP(ℕ)/Iand from automorphisms to arbitrary ho-momorphisms were made by Just in a series of papers ([14]-[17]), motivated by some problems in algebra ([7, pp. 38–39], [43, I.12.11], [45, Q48]) and topology ([46, p. 537]).

On the equivalence of certain consequences of the proper forcing axiom

AbstractWe prove that a number of axioms, each a consequence of PFA (the Proper Forcing Axiom) are equivalent. In particular we show that TOP (the Thinning-out Principle as introduced by Baumgartner in the Handbook of set-theoretic topology), is equivalent to the following statement: If I is an ideal on ω1 with ω1 generators, then there exists an uncountable X ⊆ ω1, such that either [X]ω ∩ I = ∅ or [X]ω ⊆ I.

The bounded proper forcing axiom

AbstractThe bounded proper forcing axiom BPFA is the statement that for any family of ℵ1 many maximal antichains of a proper forcing notion, each of size ℵ1, there is a directed set meeting all these antichains.A regular cardinal κ is called ∑1-reflecting, if for any regular cardinal χ, for all formulas φ, “H(χ) ⊨ ‘φ’” implies “∃δ < κ, H(δ) ⊨ ‘φ’”.We investigate several algebraic consequences of BPFA, and we show that the consistency strength of the bounded proper forcing axiom is exactly the existence of a ∑1-reflecting cardinal (which is less than the existence of a Mahlo cardinal).We also show that the question of the existence of isomorphisms between two structures can be reduced to the question of rigidity of a structure.

Forcing axioms and stationary sets

is stationary. In general, if X is any class of posets, the forcing axioms XFA and XFA + are obtained by replacing “semi-proper LY” by “9 E X” in the above definitions. Sometimes we denote these axioms by FA(X) and FA+(.X). Thus, for example, FA(ccc) is the familiar Martin’s Axiom, MAN,. The notion of semi-proper, as an extension of a previous notion of proper, was defined and extensively studied by Shelah in [Shl J. These forcing notions generalize ccc and countably closed forcing and can be iterated without collapsing N, . SPFA is implicit in [Shl] as an obvious strengthening of the Proper Forcing Axiom (PFA), previously formulated and proved consistent by Baumgartner (see [Bal, Del]). PFA had settled many combinatorial questions on the uncountable and SPFA was not

TheL <ω-theory of the class of Archimedian real closed fields

For the classA of uncountable Archimedian real closed fields we show that the statement “TheL <ω-theory ofA is complete” is independent of ZFC. In particular we have the following results: Assuming the Continuum-Hypothesis (CH) is incomplete. Conversely it is possible to build a model of set theory in which is complete and decidable. The latter can also be deduced from the Proper Forcing Axiom (PFA). In this case turns out to be equivalent to the elementary theory of the real numbers ℝ (by a quantifier-elimination procedure). Formally: is incomplete. is complete and decidable.