scholarly journals DEPENDENT CHOICE, PROPERNESS, AND GENERIC ABSOLUTENESS

2020 ◽  
pp. 1-25
Author(s):  
DAVID ASPERÓ ◽  
ASAF KARAGILA
Keyword(s):  
Large Cardinals ◽  
Proper Forcing Axiom ◽  
Generic Absoluteness ◽  
Classical Mathematics ◽  
Proper Forcing ◽  
Natural Foundation ◽  
The Right ◽  
Dependent Choice ◽  
Chang Model ◽  
Forcing Axiom

Abstract We show that Dependent Choice is a sufficient choice principle for developing the basic theory of proper forcing, and for deriving generic absoluteness for the Chang model in the presence of large cardinals, even with respect to $\mathsf {DC}$ -preserving symmetric submodels of forcing extensions. Hence, $\mathsf {ZF}+\mathsf {DC}$ not only provides the right framework for developing classical analysis, but is also the right base theory over which to safeguard truth in analysis from the independence phenomenon in the presence of large cardinals. We also investigate some basic consequences of the Proper Forcing Axiom in $\mathsf {ZF}$ , and formulate a natural question about the generic absoluteness of the Proper Forcing Axiom in $\mathsf {ZF}+\mathsf {DC}$ and $\mathsf {ZFC}$ . Our results confirm $\mathsf {ZF} + \mathsf {DC}$ as a natural foundation for a significant portion of “classical mathematics” and provide support to the idea of this theory being also a natural foundation for a large part of set theory.

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.

Forcing Axioms, Supercompact Cardinals, Singular Cardinal Combinatorics

2008 ◽  
Vol 14 (1) ◽  
pp. 99-113
Author(s):  
Matteo Viale
Keyword(s):  
Large Cardinals ◽  
Singular Cardinal ◽  
Strongly Compact Cardinal ◽  
Compact Cardinal ◽  
Forcing Axioms ◽  
Supercompact Cardinals ◽  
Saturation Properties ◽  
Proper Forcing Axiom ◽  
Singular Cardinals ◽  
Proper Forcing

The purpose of this communication is to present some recent advances on the consequences that forcing axioms and large cardinals have on the combinatorics of singular cardinals. I will introduce a few examples of problems in singular cardinal combinatorics which can be fruitfully attacked using ideas and techniques coming from the theory of forcing axioms and then translate the results so obtained in suitable large cardinals properties.The first example I will treat is the proof that the proper forcing axiom PFA implies the singular cardinal hypothesis SCH, this will easily lead to a new proof of Solovay's theorem that SCH holds above a strongly compact cardinal. I will also outline how some of the ideas involved in these proofs can be used as means to evaluate the “saturation” properties of models of strong forcing axioms like MM or PFA.The second example aims to show that the transfer principle (ℵω+1, ℵω) ↠ (ℵ2, ℵ1) fails assuming Martin's Maximum MM. Also in this case the result can be translated in a large cardinal property, however this requires a familiarity with a rather large fragment of Shelah's pcf-theory.Only sketchy arguments will be given, the reader is referred to the forthcoming [25] and [38] for a thorough analysis of these problems and for detailed proofs.

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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