scholarly journals The bounded proper forcing axiom

1995 ◽  
Vol 60 (1) ◽  
pp. 58-73 ◽  
Author(s):  
Martin Goldstern ◽  
Saharon Shelah
Keyword(s):  
Regular Cardinal ◽  
Consistency Strength ◽  
Mahlo Cardinal ◽  
Proper Forcing Axiom ◽  
Proper Forcing ◽  
Directed Set ◽  
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.

scholarly journals On the consistency strength of the proper forcing axiom

2011 ◽  
Vol 228 (5) ◽  
pp. 2672-2687 ◽  
Author(s):  
Matteo Viale ◽  
Christoph Weiß
Keyword(s):  
Consistency Strength ◽  
Proper Forcing Axiom ◽  
Proper Forcing ◽  
Forcing Axiom

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.

Hierarchies of forcing axioms II

2008 ◽  
Vol 73 (2) ◽  
pp. 522-542 ◽  
Author(s):  
Itay Neeman
Keyword(s):  
Structural Model ◽  
Free Variable ◽  
Forcing Axioms ◽  
First Order ◽  
Proper Forcing Axiom ◽  
Proper Forcing ◽  
Forcing Axiom

AbstractA truth for λ is a pair 〈Q, ψ〉 so that Q ⊆ Hλ, ψ is a first order formula with one free variable, and there exists B ⊆ Hλ+ such that (Hλ+; ∈, B) ⊨ ψ[Q]. A cardinal λ is , indescribable just in case that for every truth 〈Q, ψ〈 for λ, there exists < λ so that is a cardinal and 〈Q ∩ , ψ) is a truth for . More generally, an interval of cardinals [κ, λ] with κ ≤ λ is indescribable if for every truth 〈Q, ψ〈 for λ, there exists , and π: → Hλ so that is a cardinal, is a truth for , and π is elementary from () into (H; ∈, κ, Q) with id.We prove that the restriction of the proper forcing axiom to ϲ-linked posets requires a indescribable cardinal in L, and that the restriction of the proper forcing axiom to ϲ+-linked posets, in a proper forcing extension of a fine structural model, requires a indescribable 1-gap [κ, κ+]. These results show that the respective forward directions obtained in Hierarchies of Forcing Axioms I by Neeman and Schimmerling are optimal.

scholarly journals SET MAPPING REFLECTION

2005 ◽  
Vol 05 (01) ◽  
pp. 87-97 ◽  
Author(s):  
JUSTIN TATCH MOORE
Keyword(s):  
Reflection Principle ◽  
Axiom Of Choice ◽  
Proper Forcing Axiom ◽  
Proper Forcing ◽  
The Axiom Of Choice ◽  
Bounded Form ◽  
Forcing Axiom

In this note we will discuss a new reflection principle which follows from the Proper Forcing Axiom. The immediate purpose will be to prove that the bounded form of the Proper Forcing Axiom implies both that 2ω = ω2 and that [Formula: see text] satisfies the Axiom of Choice. It will also be demonstrated that this reflection principle implies that □(κ) fails for all regular κ > ω1.

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