Properties of forcing preserved by finite support iterations

Annals of Mathematical Logic, 1980

The Journal of Symbolic Logic, 1995

We 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 co, with co, generators, then there exists an uncountable X C co,, such that either [X]w n I = 0 or [X]w C I. ?1. Introduction. In this paper we study relations between some consequences of the Proper Forcing Axiom (PFA). Among them we consider the Thinning-out Principle (TOP) introduced by Baumgartner in [B], and the partition calculus axiom co, ) (co, (co,; fin cw1))2 proposed by Todorcevic in [T]. We show that each of these two axioms can be restated in a simpler way, and then we easily deduce that Todorcevic's axiom (which we call Axiom S in this paper) is a consequence of TOP. We will then show how our versions of these axioms give simplified proofs of the applications of these axioms in [B] and [T]. We will show that the following axiom is equivalent to TOP: AXIOM 0. Let S = {S,: a < co } be a collection of (countable) subsets of coi, such that for every uncountable X C co,, there exists a countable set Q C X which cannot be covered by a finite subfamily of S. Then there exists an uncountable subset of co1 which meets each S, in a finite set. We will show that the following (weaker) version of Axiom 0 is equivalent to Axiom S: AXIOM 1. Let S = {S,: a < co, } be a collection of (countable) subsets of W1, such that for every uncountable X C co,, there exists a countable set Q C X which cannot be covered by a finite subfamily of S. Then there exists uncountable set X C co,, such that for each a C X, S, n X is a finite set.

Annals of Pure and Applied Logic, 2013

We study the spectrum of forcing notions between the iterations of-closed followed by ccc forcings and the proper forcings. This includes the hierarchy of ↵-proper forcings for indecomposable countable ordinals ↵, the Axiom A forcings and forcings completely embeddable into an iteration of a-closed followed by a ccc forcing. For the latter class, we present an equivalent characterization in terms of Baumgartner's Axiom A. This resolves a conjecture of Baumgartner from the 1980s. We also study the bounded forcing axioms for the hierarchy of ↵proper forcings. Following ideas of Shelah we separate them for distinct countable indecomposable ordinals.

Discrete Mathematics, 1983

Let P be a ranked partially ordered set. An h-family is a subset of P such that no h + 1 elements of the family lie on any single chain. P has the strong h-family property, if each maximal h-family in P is the union of h complete levels. Sufficient conditions for the strong h-family property are given.

Notre Dame Journal of Formal Logic - NDJFL, 2005

An outline is given of the proof that the consistency of a κ⁺-Mahlo cardinal implies that of the statement that I[ω₂] does not include any stationary subsets of Cof(ω₁). An additional discussion of the techniques of this proof includes their use to obtain a model with no ω₂-Aronszajn tree and to add an ω₂-Souslin tree with finite conditions.

Lecture Notes in Computer Science, 2013

We survey some classical and some recent results in the theory of forcing axioms, aiming to present recent breakthroughs and interest the reader in further developing the theory. The article is written for an audience of logicians and mathematicians not necessarily familiar with set theory.

Archive for Mathematical Logic, 2000

We show that Bounded Forcing Axioms (for instance, Martin's Axiom, the Bounded Proper Forcing Axiom, or the Bounded Martin's Maximum) are equivalent to principles of generic absoluteness, that is, they assert that if a Σ 1 sentence of the language of set theory with parameters of small transitive size is forceable, then it is true. We also show that Bounded Forcing Axioms imply a strong form of generic absoluteness for projective sentences, namely, if a Σ 1 3 sentence with parameters is forceable, then it is true. Further, if for every real x, x exists, and the second uniform indiscernible is less than ω 2 , then the same holds for Σ 1 4 sentences.

1998

We present a systematic study of the method of &quot;norms on possibilities&quot; 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&#x27;&#x27; (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.

The Journal of Symbolic Logic, 2012

We present some abstract theorems showing how domination properties equivalent to being or array nonrecursive can be used to construct sets generic for different notions of forcing. These theorems are then applied to give simple proofs of some known results. We also give a direct uniform proof of a recent result of Ambos-Spies, Ding, Wang, and Yu [2009] that every degree above any in is recursively enumerable in a 1-generic degree strictly below it. Our major new result is that every array nonrecursive degree is r.e. in some degree strictly below it. Our analysis of array nonrecursiveness and construction of generic sequences below ANR degrees also reveal a new level of uniformity in these types of results.