Indestructibility of some compactness principles over models of PFA

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F24%3A00576355" target="_blank" >RIV/67985840:_____/24:00576355 - isvavai.cz</a>

Nalezeny alternativní kódy

RIV/00216208:11210/24:10477924

Výsledek na webu

<a href="https://doi.org/10.1016/j.apal.2023.103359" target="_blank" >https://doi.org/10.1016/j.apal.2023.103359</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1016/j.apal.2023.103359" target="_blank" >10.1016/j.apal.2023.103359</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Indestructibility of some compactness principles over models of PFA

Popis výsledku v původním jazyce

We show that PFA (Proper Forcing Axiom) implies that adding any number of Cohen subsets of ω will not add an ω2-Aronszajn tree or a weak ω1-Kurepa tree, and moreover no σ-centered forcing can add a weak ω1-Kurepa tree (a tree of height and size ω1 with at least ω2 cofinal branches). This partially answers an open problem whether ccc forcings can add ω2-Aronszajn trees or ω1-Kurepa trees (with ¬□ω in the latter case). We actually prove more: We show that a consequence of PFA, namely the guessing model principle, GMP, which is equivalent to the ineffable slender tree property, ISP, is preserved by adding any number of Cohen subsets of ω. And moreover, GMP implies that no σ-centered forcing can add a weak ω1-Kurepa tree (see Section 2.1 for definitions). For more generality, we study variations of the principle GMP at higher cardinals and the indestructibility consequences they entail, and as applications we answer a question of Mohammadpour about guessing models at weakly but not strongly inaccessible cardinals and show that there is a model in which there are no weak ℵω+1-Kurepa trees and no ℵω+2-Aronszajn trees.

Název v anglickém jazyce

Indestructibility of some compactness principles over models of PFA

Popis výsledku anglicky

We show that PFA (Proper Forcing Axiom) implies that adding any number of Cohen subsets of ω will not add an ω2-Aronszajn tree or a weak ω1-Kurepa tree, and moreover no σ-centered forcing can add a weak ω1-Kurepa tree (a tree of height and size ω1 with at least ω2 cofinal branches). This partially answers an open problem whether ccc forcings can add ω2-Aronszajn trees or ω1-Kurepa trees (with ¬□ω in the latter case). We actually prove more: We show that a consequence of PFA, namely the guessing model principle, GMP, which is equivalent to the ineffable slender tree property, ISP, is preserved by adding any number of Cohen subsets of ω. And moreover, GMP implies that no σ-centered forcing can add a weak ω1-Kurepa tree (see Section 2.1 for definitions). For more generality, we study variations of the principle GMP at higher cardinals and the indestructibility consequences they entail, and as applications we answer a question of Mohammadpour about guessing models at weakly but not strongly inaccessible cardinals and show that there is a model in which there are no weak ℵω+1-Kurepa trees and no ℵω+2-Aronszajn trees.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GF19-29633L" target="_blank" >GF19-29633L: Kompaktnostní principy a kombinatorické vlastnosti</a><br>

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2024

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Annals of Pure and Applied Logic

ISSN

0168-0072

e-ISSN

1873-2461

Svazek periodika

175

Číslo periodika v rámci svazku

1

Stát vydavatele periodika

NL - Nizozemsko

Počet stran výsledku

17

Strana od-do

103359

Kód UT WoS článku

001078824100001

EID výsledku v databázi Scopus

2-s2.0-85171791208