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 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.

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.

Woodin's axiom (*), bounded forcing axioms, and precipitous ideals on ω1

2012 ◽  
Vol 77 (2) ◽  
pp. 475-498 ◽  
Author(s):  
Benjamin Claverie ◽  
Ralf Schindler
Keyword(s):  
Model Operator ◽  
Forcing Axioms ◽  
Woodin Cardinals ◽  
Proper Forcing Axiom ◽  
Proper Forcing ◽  
Reflection Principles ◽  
Precipitous Ideals ◽  
Forcing Axiom ◽  
The Way

AbstractIf the Bounded Proper Forcing Axiom BPFA holds, then Mouse Reflection holds at ℵ2 with respect to all mouse operators up to the level of Woodin cardinals in the next ZFC-model. This yields that if Woodin's ℙmax axiom (*) holds, then BPFA implies that V is closed under the “Woodin-in-the-next-ZFC-model” operator. We also discuss stronger Mouse Reflection principles which we show to follow from strengthenings of BPFA, and we discuss the theory BPFA plus “NSω1 is precipitous” and strengthenings thereof. Along the way, we answer a question of Baumgartner and Taylor, [2, Question 6.11].

Completely Additive Liftings

1998 ◽  
Vol 4 (1) ◽  
pp. 37-54 ◽  
Author(s):  
Ilijas Farah
Keyword(s):  
Set Theory ◽  
Classical Result ◽  
Continuum Hypothesis ◽  
Axiomatic Approach ◽  
Proper Forcing Axiom ◽  
And Topology ◽  
Proper Forcing ◽  
Forcing Axiom ◽  
The Continuum ◽  
The Ideal

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 of P(ℕ)/ Fin, where Fin is the ideal of finite sets, is trivial, 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 of P(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 quotient P(ℕ)/ Fin to quotients over more general ideals P(ℕ)/I and 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]).

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