scholarly journals The proper forcing axiom and the singular cardinal hypothesis

2006 ◽  
Vol 71 (2) ◽  
pp. 473-479 ◽  
Author(s):  
Matteo Viale
Keyword(s):  
Reflection Principle ◽  
Singular Cardinal ◽  
Proper Forcing Axiom ◽  
Singular Cardinal Hypothesis ◽  
Proper Forcing ◽  
Forcing Axiom

AbstractWe show that the Proper Forcing Axiom implies the Singular Cardinal Hypothesis. The proof uses the reflection principle MRP introduced by Moore in [11].

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.

PFA implies ADL(ℝ)

2005 ◽  
Vol 70 (4) ◽  
pp. 1255-1296 ◽  
Author(s):  
John R. Steel
Keyword(s):  
Reflection Principle ◽  
Inaccessible Cardinal ◽  
Strong Limit ◽  
Proper Forcing Axiom ◽  
Inner Model ◽  
Strong Reflection ◽  
Logical Strength ◽  
Proper Forcing ◽  
Uncountable Cardinals ◽  
Forcing Axiom

In this paper we shall proveTheorem 0.1. Suppose there is a singular strong limit cardinal κ such that □κ fails; then AD holds in L(R).See [10] for a discussion of the background to this problem. We suspect that more work will produce a proof of the theorem with its hypothesis that κ is a strong limit weakened to ∀α < κ (αω < κ), and significantly more work will enable one to drop the hypothesis that K is a strong limit entirely. At present, we do not see how to carry out even the less ambitious project.Todorcevic [23] has shown that if the Proper Forcing Axiom (PFA) holds, then □κ fails for all uncountable cardinals κ. Thus we get immediately:It has been known since the early 90's that PFA implies PD, that PFA plus the existence of a strongly inaccessible cardinal implies ADL(ℝ) and that PFA plus a measurable yields an inner model of ADℝ containing all reals and ordinals. As we do here, these arguments made use of Tororcevic's work, so that logical strength is ultimately coming from a failure of covering for some appropriate core models.In late 2000, A. S. Zoble and the author showed that (certain consequences of) Todorcevic's Strong Reflection Principle (SRP) imply ADL(ℝ). (See [22].) Since Martin's Maximum implies SRP, this gave the first derivation of ADL(ℝ) from an “unaugmented” 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.

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.

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