Embedding orders into the cardinals with $D{C}_{\kappa }$

Asaf Karagila. "Embedding orders into the cardinals with $DC_{κ}$." Fundamenta Mathematicae 226.2 (2014): 143-156. <http://eudml.org/doc/283313>.

@article{AsafKaragila2014,
abstract = {Jech proved that every partially ordered set can be embedded into the cardinals of some model of ZF. We extend this result to show that every partially ordered set can be embedded into the cardinals of some model of $ZF + DC_\{<κ\}$ for any regular κ. We use this theorem to show that for all κ, the assumption of $DC_\{κ\}$ does not entail that there are no decreasing chains of cardinals. We also show how to extend the result to and embed into the cardinals a proper class which is definable over the ground model. We use this extension to give a large-cardinals-free proof of independence of the weak choice principle known as WISC from $DC_\{κ\}$.},
author = {Asaf Karagila},
journal = {Fundamenta Mathematicae},
keywords = {axiom of choice; symmetric extensions; cardinals},
language = {eng},
number = {2},
pages = {143-156},
title = {Embedding orders into the cardinals with $DC_\{κ\}$},
url = {http://eudml.org/doc/283313},
volume = {226},
year = {2014},
}

TY - JOUR
AU - Asaf Karagila
TI - Embedding orders into the cardinals with $DC_{κ}$
JO - Fundamenta Mathematicae
PY - 2014
VL - 226
IS - 2
SP - 143
EP - 156
AB - Jech proved that every partially ordered set can be embedded into the cardinals of some model of ZF. We extend this result to show that every partially ordered set can be embedded into the cardinals of some model of $ZF + DC_{<κ}$ for any regular κ. We use this theorem to show that for all κ, the assumption of $DC_{κ}$ does not entail that there are no decreasing chains of cardinals. We also show how to extend the result to and embed into the cardinals a proper class which is definable over the ground model. We use this extension to give a large-cardinals-free proof of independence of the weak choice principle known as WISC from $DC_{κ}$.
LA - eng
KW - axiom of choice; symmetric extensions; cardinals
UR - http://eudml.org/doc/283313
ER -