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 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 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.

Iterated reflection principles and the ω-rule

1982 ◽  
Vol 47 (4) ◽  
pp. 721-733 ◽  
Author(s):  
Ulf R. Schmerl
Keyword(s):  
Reflection Principle ◽  
Peano Arithmetic ◽  
Equivalent System ◽  
Formal Similarity ◽  
Sound System ◽  
Formal Systems ◽  
Ordinal Notations ◽  
Axiomatizable Theory ◽  
Reflection Principles ◽  
Image Position

The ω-rule,with the meaning “if the formula A(n) is provable for all n, then the formula ∀xA(x) is provable”, has a certain formal similarity with a uniform reflection principle saying “if A(n) is provable for all n, then ∀xA(x) is true”. There are indeed some hints in the literature that uniform reflection has sometimes been understood as a “formalized ω-rule” (cf. for example S. Feferman [1], G. Kreisel [3], G. H. Müller [7]). This similarity has even another aspect: replacing the induction rule or scheme in Peano arithmetic PA by the ω-rule leads to a complete and sound system PA∞, where each true arithmetical statement is provable. In [2] Feferman showed that an equivalent system can be obtained by erecting on PA a transfinite progression of formal systems PAα based on iterations of the uniform reflection principle according to the following scheme:Then T = (∪dЄ, PAd, being Kleene's system of ordinal notations, is equivalent to PA∞. Of course, T cannot be an axiomatizable theory.

scholarly journals Simultaneous stationary reflection and square sequences

2017 ◽  
Vol 17 (02) ◽  
pp. 1750010 ◽  
Author(s):  
Yair Hayut ◽  
Chris Lambie-Hanson
Keyword(s):  
Stationary Reflection ◽  
Square Principles ◽  
Square Sequences ◽  
The Relationship ◽  
Weak Square

We investigate the relationship between weak square principles and simultaneous reflection of stationary sets.

On ideals and stationary reflection

1989 ◽  
Vol 54 (2) ◽  
pp. 568-575 ◽  
Author(s):  
C. A. Johnson
Keyword(s):  
Stationary Reflection ◽  
Reflection Property ◽  
Regular Cardinals ◽  
The Ideal

It is a theorem of Prikry [7] that ifκcarries a uniformη-descendingly complete ultrafilter then the stationary reflection propertyfails. In this paper we will derive similar results, but here from properties of filters (or ideals) rather than ultrafilters.Throughoutκandηwill denote regular cardinals withη<κ(in particularκwill be uncountable), andIwill denote an ideal onκ, by which we mean a setI⊆P(κ) such that (i)Iis closed under taking subsets and finite unions and (ii)αЄIfor eachα<κ, butκ∉I.Iis said to beμ-complete if it is closed under taking unions of size <μ,I* = {X⊆κ∣κ−XЄI} is the filter dual toIand ifAЄI+(=P(κ) −I), thenI∣Ais the ideal onκgiven byI∣A= {X⊆κ∣X∩AЄI}. Ifh: A→κthenhis said to be (i) unbounded modIif for eachα<κ,h−1(α) = {ξЄA∣h(ξ)<α} ЄIand (ii) a least function forIifhis unbounded modIand wheneverg: A→κis a function, unbounded modI, then {ξЄA∣g{ξ) <h{ξ)} ЄI.

scholarly journals A STRONG REFLECTION PRINCIPLE

2017 ◽  
Vol 10 (4) ◽  
pp. 651-662 ◽  
Author(s):  
SAM ROBERTS
Keyword(s):  
Set Theory ◽  
Reflection Principle ◽  
Higher Order ◽  
Second Order ◽  
Proper Class ◽  
Strong Reflection ◽  
Reflection Principles ◽  
Extendible Cardinals ◽  
Order Set

AbstractThis article introduces a new reflection principle. It is based on the idea that whatever is true in all entities of some kind is also true in a set-sized collection of them. Unlike standard reflection principles, it does not re-interpret parameters or predicates. This allows it to be both consistent in all higher-order languages and remarkably strong. For example, I show that in the language of second-order set theory with predicates for a satisfaction relation, it is consistent relative to the existence of a 2-extendible cardinal (Theorem 7.12) and implies the existence of a proper class of 1-extendible cardinals (Theorem 7.9).

Separating stationary reflection principles

2000 ◽  
Vol 65 (1) ◽  
pp. 247-258 ◽  
Author(s):  
Paul Larson
Keyword(s):  
Stationary Reflection ◽  
Reflection Principles

AbstractWe present a variety of (ω, ∞)-distributive forcings which when applied to models of Martin's Maximum separate certain well known reflection principles. In particular, we do this for the reflection principles SR, SRα (α ≤ ω1), and SRP.

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