On the set-theoretic strength of the n-compactness of generalized Cantor cubes

Paul Howard, and Eleftherios Tachtsis. "On the set-theoretic strength of the n-compactness of generalized Cantor cubes." Fundamenta Mathematicae 234.3 (2016): 241-252. <http://eudml.org/doc/286366>.

@article{PaulHoward2016,
abstract = {We investigate, in set theory without the Axiom of Choice , the set-theoretic strength of the statement Q(n): For every infinite set X, the Tychonoff product $2^\{X\}$, where 2 = 0,1 has the discrete topology, is n-compact, where n = 2,3,4,5 (definitions are given in Section 1). We establish the following results: (1) For n = 3,4,5, Q(n) is, in (Zermelo-Fraenkel set theory minus ), equivalent to the Boolean Prime Ideal Theorem , whereas (2) Q(2) is strictly weaker than in set theory (Zermelo-Fraenkel set theory with the Axiom of Extensionality weakened in order to allow atoms). This settles the open problem in Tachtsis (2012) on the relation of Q(n), n = 2,3,4,5, to .},
author = {Paul Howard, Eleftherios Tachtsis},
journal = {Fundamenta Mathematicae},
keywords = {axiom of choice; Boolean prime ideal theorem; compactness and $n$-compactness of generalized Cantor cubes; Fraenkel-Mostowski (FM) permutation models},
language = {eng},
number = {3},
pages = {241-252},
title = {On the set-theoretic strength of the n-compactness of generalized Cantor cubes},
url = {http://eudml.org/doc/286366},
volume = {234},
year = {2016},
}

TY - JOUR
AU - Paul Howard
AU - Eleftherios Tachtsis
TI - On the set-theoretic strength of the n-compactness of generalized Cantor cubes
JO - Fundamenta Mathematicae
PY - 2016
VL - 234
IS - 3
SP - 241
EP - 252
AB - We investigate, in set theory without the Axiom of Choice , the set-theoretic strength of the statement Q(n): For every infinite set X, the Tychonoff product $2^{X}$, where 2 = 0,1 has the discrete topology, is n-compact, where n = 2,3,4,5 (definitions are given in Section 1). We establish the following results: (1) For n = 3,4,5, Q(n) is, in (Zermelo-Fraenkel set theory minus ), equivalent to the Boolean Prime Ideal Theorem , whereas (2) Q(2) is strictly weaker than in set theory (Zermelo-Fraenkel set theory with the Axiom of Extensionality weakened in order to allow atoms). This settles the open problem in Tachtsis (2012) on the relation of Q(n), n = 2,3,4,5, to .
LA - eng
KW - axiom of choice; Boolean prime ideal theorem; compactness and $n$-compactness of generalized Cantor cubes; Fraenkel-Mostowski (FM) permutation models
UR - http://eudml.org/doc/286366
ER -