Coloring ordinals by reals

Jörg Brendle, and Sakaé Fuchino. "Coloring ordinals by reals." Fundamenta Mathematicae 196.2 (2007): 151-195. <http://eudml.org/doc/286533>.

@article{JörgBrendle2007,
abstract = {We study combinatorial principles we call the Homogeneity Principle HP(κ) and the Injectivity Principle IP(κ,λ) for regular κ > ℵ₁ and λ ≤ κ which are formulated in terms of coloring the ordinals < κ by reals. These principles are strengthenings of $C^\{s\}(κ)$ and $F^\{s\}(κ)$ of I. Juhász, L. Soukup and Z. Szentmiklóssy. Generalizing their results, we show e.g. that IP(ℵ₂,ℵ₁) (hence also IP(ℵ₂,ℵ₂) as well as HP(ℵ₂)) holds in a generic extension of a model of CH by Cohen forcing, and IP(ℵ₂,ℵ₂) (hence also HP(ℵ₂)) holds in a generic extension by countable support side-by-side product of Sacks or Prikry-Silver forcing (Corollary 4.8). We also show that the latter result is optimal (Theorem 5.2). Relations between these principles and their influence on the values of the variations $^\{↑\}$, $^\{h\}$, *, of the bounding number are studied. One of the consequences of HP(κ) besides $C^\{s\}(κ)$ is that there is no projective well-ordering of length κ on any subset of $^\{ω\}ω$. We construct a model in which there is no projective well-ordering of length ω₂ on any subset of $^\{ω\}ω$ ( = ℵ₁ in our terminology) while * = ℵ₂ (Theorem 6.4).},
author = {Jörg Brendle, Sakaé Fuchino},
journal = {Fundamenta Mathematicae},
keywords = {Homogeneity Principle; Injectivity Principle; bounding number; projective well-ordering; Cohen forcing; Brendle-LaBerge forcing; Prikry-Silver forcing},
language = {eng},
number = {2},
pages = {151-195},
title = {Coloring ordinals by reals},
url = {http://eudml.org/doc/286533},
volume = {196},
year = {2007},
}

TY - JOUR
AU - Jörg Brendle
AU - Sakaé Fuchino
TI - Coloring ordinals by reals
JO - Fundamenta Mathematicae
PY - 2007
VL - 196
IS - 2
SP - 151
EP - 195
AB - We study combinatorial principles we call the Homogeneity Principle HP(κ) and the Injectivity Principle IP(κ,λ) for regular κ > ℵ₁ and λ ≤ κ which are formulated in terms of coloring the ordinals < κ by reals. These principles are strengthenings of $C^{s}(κ)$ and $F^{s}(κ)$ of I. Juhász, L. Soukup and Z. Szentmiklóssy. Generalizing their results, we show e.g. that IP(ℵ₂,ℵ₁) (hence also IP(ℵ₂,ℵ₂) as well as HP(ℵ₂)) holds in a generic extension of a model of CH by Cohen forcing, and IP(ℵ₂,ℵ₂) (hence also HP(ℵ₂)) holds in a generic extension by countable support side-by-side product of Sacks or Prikry-Silver forcing (Corollary 4.8). We also show that the latter result is optimal (Theorem 5.2). Relations between these principles and their influence on the values of the variations $^{↑}$, $^{h}$, *, of the bounding number are studied. One of the consequences of HP(κ) besides $C^{s}(κ)$ is that there is no projective well-ordering of length κ on any subset of $^{ω}ω$. We construct a model in which there is no projective well-ordering of length ω₂ on any subset of $^{ω}ω$ ( = ℵ₁ in our terminology) while * = ℵ₂ (Theorem 6.4).
LA - eng
KW - Homogeneity Principle; Injectivity Principle; bounding number; projective well-ordering; Cohen forcing; Brendle-LaBerge forcing; Prikry-Silver forcing
UR - http://eudml.org/doc/286533
ER -