scholarly journals Stationary reflection principles and two cardinal tree properties

2013 ◽  
Vol 14 (1) ◽  
pp. 69-85 ◽  
Author(s):  
Hiroshi Sakai ◽  
Boban Veličković
Keyword(s):  
Weak Form ◽  
Reflection Principle ◽  
Singular Cardinal ◽  
Stationary Reflection ◽  
Strong Compactness ◽  
Singular Cardinal Hypothesis ◽  
Martin’S Axiom ◽  
Reflection Principles ◽  
Stationary Set ◽  
Weak Square

AbstractWe study the consequences of stationary and semi-stationary set reflection. We show that the semi-stationary reflection principle implies the Singular Cardinal Hypothesis, the failure of the weak square principle, etc. We also consider two cardinal tree properties introduced recently by Weiss, and prove that they follow from stationary and semi-stationary set reflection augmented with a weak form of Martin’s Axiom. We also show that there are some differences between the two reflection principles, which suggests that stationary set reflection is analogous to supercompactness, whereas semi-stationary set reflection is analogous to strong compactness.

scholarly journals Rado’s Conjecture and its Baire version

2019 ◽  
Vol 20 (01) ◽  
pp. 1950015
Author(s):  
Jing Zhang
Keyword(s):  
Reflection Principle ◽  
Stationary Reflection ◽  
Forcing Axioms ◽  
Martin’S Axiom ◽  
Partition Relations ◽  
Consistency Results ◽  
Square Principles ◽  
Reflection Principles ◽  
Weak Square ◽  
Forcing Axiom

Rado’s Conjecture is a compactness/reflection principle that says any nonspecial tree of height [Formula: see text] has a nonspecial subtree of size [Formula: see text]. Though incompatible with Martin’s Axiom, Rado’s Conjecture turns out to have many interesting consequences that are also implied by certain forcing axioms. In this paper, we obtain consistency results concerning Rado’s Conjecture and its Baire version. In particular, we show that a fragment of [Formula: see text], which is the forcing axiom for Baire Indestructibly Proper forcings, is compatible with the Baire Rado’s Conjecture. As a corollary, the Baire Rado’s Conjecture does not imply Rado’s Conjecture. Then we discuss the strength and limitations of the Baire Rado’s Conjecture regarding its interaction with stationary reflection principles and some families of weak square principles. Finally, we investigate the influence of Rado’s Conjecture on some polarized partition relations.

scholarly journals Sigma-Prikry forcing II: Iteration Scheme

2021 ◽  
pp. 2150019
Author(s):  
Alejandro Poveda ◽  
Assaf Rinot ◽  
Dima Sinapova
Keyword(s):  
Blow Up ◽  
General Scheme ◽  
Iteration Scheme ◽  
Singular Cardinal ◽  
Prikry Forcing ◽  
Supercompact Cardinals ◽  
Singular Cardinal Hypothesis ◽  
Singular Cardinals ◽  
Stationary Set ◽  
Stationary Subsets

In Part I of this series [5], we introduced a class of notions of forcing which we call [Formula: see text]-Prikry, and showed that many of the known Prikry-type notions of forcing that centers around singular cardinals of countable cofinality are [Formula: see text]-Prikry. We proved that given a [Formula: see text]-Prikry poset [Formula: see text] and a [Formula: see text]-name for a nonreflecting stationary set [Formula: see text], there exists a corresponding [Formula: see text]-Prikry poset that projects to [Formula: see text] and kills the stationarity of [Formula: see text]. In this paper, we develop a general scheme for iterating [Formula: see text]-Prikry posets, as well as verify that the Extender-based Prikry forcing is [Formula: see text]-Prikry. As an application, we blow-up the power of a countable limit of Laver-indestructible supercompact cardinals, and then iteratively kill all nonreflecting stationary subsets of its successor. This yields a model in which the singular cardinal hypothesis fails and simultaneous reflection of finite families of stationary sets holds.

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].

Strong analogues of Martin's axiom Imply Axiom R

1987 ◽  
Vol 52 (1) ◽  
pp. 216-218 ◽  
Author(s):  
Robert E. Beaudoin
Keyword(s):  
Reflection Principle ◽  
Partial Orders ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
Stationary Set ◽  
Stationary Subsets

AbstractWe show that either PFA+ or Martin's maximum implies Fleissner's Axiom R, a reflection principle for stationary subsets of Pℵ(λ). In fact, the “plus version” (for one term denoting a stationary set) of Martin's axiom for countably closed partial orders implies Axiom R.

scholarly journals Guessing models and the approachability ideal

2020 ◽  
pp. 2150003
Author(s):  
Rahman Mohammadpour ◽  
Boban Veličković
Keyword(s):  
Generic Extension ◽  
Singular Cardinal ◽  
Tree Property ◽  
Supercompact Cardinals ◽  
Ideal Formula ◽  
Singular Cardinal Hypothesis ◽  
Nonstationary Ideal ◽  
The Universe ◽  
Weak Square

Starting with two supercompact cardinals we produce a generic extension of the universe in which a principle that we call [Formula: see text] holds. This principle implies [Formula: see text] and [Formula: see text], and hence the tree property at [Formula: see text] and [Formula: see text], the Singular Cardinal Hypothesis, and the failure of the weak square principle [Formula: see text], for all regular [Formula: see text]. In addition, it implies that the restriction of the approachability ideal [Formula: see text] to the set of ordinals of cofinality [Formula: see text] is the nonstationary ideal on this set. The consistency of this last statement was previously shown by W. Mitchell.

scholarly journals Semistationary and stationary reflection

2008 ◽  
Vol 73 (1) ◽  
pp. 181-192 ◽  
Author(s):  
Hiroshi Sakai
Keyword(s):  
Reflection Principle ◽  
Stationary Subset ◽  
Stationary Reflection ◽  
Regular Cardinals ◽  
Reflection Principles ◽  
The Relationship

AbstractWe study the relationship between the semistationary reflection principle and stationary reflection principles. We show that for all regular cardinals λ ≥ ω2 the semistationary reflection principle in the space [λ]ω implies that every stationary subset of ≔ {α ∈ λ ∣ cf(α) = ω} reflects. We also show that for all cardinals λ ≥ ω3 the semistationary reflection principle in [λ]ω does not imply the stationary reflection principle in [λ]ω.

scholarly journals The tree property and the failure of the Singular Cardinal Hypothesis at ℵω2

2012 ◽  
Vol 77 (3) ◽  
pp. 934-946 ◽  
Author(s):  
Dima Sinapova
Keyword(s):  
Generic Extension ◽  
Singular Cardinal ◽  
Tree Property ◽  
Supercompact Cardinals ◽  
Singular Cardinal Hypothesis

AbstractWe show that given ω many supercompact cardinals, there is a generic extension in which the tree property holds at ℵω2+ 1 and the SCH fails at ℵω2.

scholarly journals MODAL STRUCTURALISM AND REFLECTION

2018 ◽  
Vol 12 (4) ◽  
pp. 823-860 ◽  
Author(s):  
SAM ROBERTS
Keyword(s):  
Set Theory ◽  
Reflection Principle ◽  
Reflection Principles

AbstractModal structuralism promises an interpretation of set theory that avoids commitment to abstracta. This article investigates its underlying assumptions. In the first part, I start by highlighting some shortcomings of the standard axiomatisation of modal structuralism, and propose a new axiomatisation I call MSST (for Modal Structural Set Theory). The main theorem is that MSST interprets exactly Zermelo set theory plus the claim that every set is in some inaccessible rank of the cumulative hierarchy. In the second part of the article, I look at the prospects for supplementing MSST with a modal structural reflection principle, as suggested in Hellman (2015). I show that Hellman’s principle is inconsistent (Theorem 5.32), and argue that modal structural reflection principles in general are either incompatible with modal structuralism or extremely weak.

A definable failure of the singular cardinal hypothesis

2012 ◽  
Vol 192 (2) ◽  
pp. 719-762 ◽  
Author(s):  
Sy-David Friedman ◽  
Radek Honzik
Keyword(s):  
Singular Cardinal ◽  
Singular Cardinal Hypothesis
Sign in / Sign up

Export Citation Format

Share Document