Partial choice functions for families of finite sets

Eric J. Hall, and Saharon Shelah. "Partial choice functions for families of finite sets." Fundamenta Mathematicae 220.3 (2013): 207-216. <http://eudml.org/doc/286109>.

@article{EricJ2013,
abstract = {Let m ≥ 2 be an integer. We show that ZF + “Every countable set of m-element sets has an infinite partial choice function” is not strong enough to prove that every countable set of m-element sets has a choice function, answering an open question from . (Actually a slightly stronger result is obtained.) The independence result in the case where m = p is prime is obtained by way of a permutation (Fraenkel-Mostowski) model of ZFA, in which the set of atoms (urelements) has the structure of a vector space over the finite field $_\{p\}$. The use of atoms is then eliminated by citing an embedding theorem of Pincus. In the case where m is not prime, suitable permutation models are built from the models used in the prime cases.},
author = {Eric J. Hall, Saharon Shelah},
journal = {Fundamenta Mathematicae},
keywords = {choice function; permutation model},
language = {eng},
number = {3},
pages = {207-216},
title = {Partial choice functions for families of finite sets},
url = {http://eudml.org/doc/286109},
volume = {220},
year = {2013},
}

TY - JOUR
AU - Eric J. Hall
AU - Saharon Shelah
TI - Partial choice functions for families of finite sets
JO - Fundamenta Mathematicae
PY - 2013
VL - 220
IS - 3
SP - 207
EP - 216
AB - Let m ≥ 2 be an integer. We show that ZF + “Every countable set of m-element sets has an infinite partial choice function” is not strong enough to prove that every countable set of m-element sets has a choice function, answering an open question from . (Actually a slightly stronger result is obtained.) The independence result in the case where m = p is prime is obtained by way of a permutation (Fraenkel-Mostowski) model of ZFA, in which the set of atoms (urelements) has the structure of a vector space over the finite field $_{p}$. The use of atoms is then eliminated by citing an embedding theorem of Pincus. In the case where m is not prime, suitable permutation models are built from the models used in the prime cases.
LA - eng
KW - choice function; permutation model
UR - http://eudml.org/doc/286109
ER -