PRESERVATION OF SUSLIN TREES AND SIDE CONDITIONS

2016 ◽  
Vol 81 (2) ◽  
pp. 483-492
Author(s):  
GIORGIO VENTURI
Keyword(s):  
Countable Support ◽  
Suslin Tree ◽  
Preservation Theorem ◽  
Proper Forcing ◽  
Countable Support Iteration ◽  
Forcing Axiom

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.

On iterating semiproper preorders

2002 ◽  
Vol 67 (4) ◽  
pp. 1431-1468 ◽  
Author(s):  
Tadatoshi Miyamoto
Keyword(s):  
Countable Support ◽  
Arbitrary Length ◽  
Countable Support Iteration ◽  
Forcing Axiom

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.

More on proper forcing

1984 ◽  
Vol 49 (4) ◽  
pp. 1034-1038 ◽  
Author(s):  
Saharon Shelah
Keyword(s):  
Dense Subset ◽  
Countable Support ◽  
Unbounded Subset ◽  
Proper Forcing ◽  
Countable Support Iteration ◽  
Closed Unbounded Subset

§1. A counterexample and preservation of “proper + X”.Theorem. Suppose V satisfies, , and for some A ⊆ ω1, every B ⊆ ω1, belongs to L[A].Then we can define a countable support iterationsuch that the following conditions hold:a) EachQiis proper and ⊩Pi “Qi, has power ℵ1”.b) Each Qi is -complete for some simple ℵ1-completeness system.c) Forcing with Pα = Lim adds reals.Proof. We shall define Qi by induction on i so that conditions a) and b) are satisfied, and Ci, is a Qi-name of a closed unbounded subset of ω1. Let : ξ < ω1› ∈ L[A] be a list of all functions f which are from δ to δ for some δ < ω1 and let h: ω1 → ω1, h ∈ L[A], be defined by h(α) = Min{β: β > α and Lβ[A]⊨ “∣α∣ = ℵ0”}.Suppose we have defined Qj for every j < i; then Pi is defined, is proper (as each Qj, j < i, is proper, and by III 3.2) and has a dense subset of power ℵ (by III 4.1). Let Gi ⊆ Pi be generic so clearly there is B ⊆ ω1, such that in V[Gi] every subset of ω1 belongs to L[A, B], The following now follows:Fact. In V[Gi], every countableN ⥽(H(ℵ2), ∈, A, B) is isomorphic toLβ[A ∩ δ, B ∩ δ] for some β < h(δ), where δ = δ(N) = ω1, ∩ N.

scholarly journals On the equivalence of certain consequences of the proper forcing axiom

1995 ◽  
Vol 60 (2) ◽  
pp. 431-443 ◽  
Author(s):  
Peter Nyikos ◽  
Leszek Piątkiewicz
Keyword(s):  
Proper Forcing Axiom ◽  
Proper Forcing ◽  
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.

BPFA and projective well-orderings of the reals

2011 ◽  
Vol 76 (4) ◽  
pp. 1126-1136 ◽  
Author(s):  
Andrés Eduardo Caicedo ◽  
Sy-David Friedman
Keyword(s):  
Coding Scheme ◽  
Proper Forcing Axiom ◽  
Proper Forcing ◽  
Forcing Axiom

AbstractIf the bounded proper forcing axiom BPFA holds and ω1 = ω1L, then there is a lightface Σ31 well-ordering of the reals. The argument combines a well-ordering due to Caicedo-Veličković with an absoluteness result for models of MA in the spirit of “David's trick.” We also present a general coding scheme that allows us to show that BPFA is equiconsistent with R being lightface Σ41 for many “consistently locally certified” relations R on ℝ. This is accomplished through a use of David's trick and a coding through the Σ2 stable ordinals of L.

scholarly journals Scott's problem for Proper Scott sets

2008 ◽  
Vol 73 (3) ◽  
pp. 845-860 ◽  
Author(s):  
Victoria Gitman
Keyword(s):  
Boolean Algebra ◽  
Partial Order ◽  
Standard System ◽  
Proper Forcing Axiom ◽  
Proper Forcing ◽  
Scott Set ◽  
Forcing Axiom

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.

The countable support iteration ideals

2013 ◽  
pp. 244-263
Author(s):  
Vladimir Kanovei ◽  
Marcin Sabok ◽  
Jindrich Zapletal
Keyword(s):  
Countable Support ◽  
Countable Support Iteration
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