On certain non-constructive properties of infinite-dimensional vector spaces

@article{Tachtsis2018,
abstract = {In set theory without the axiom of choice ($\{\rm AC\}$), we study certain non-constructive properties of infinite-dimensional vector spaces. Among several results, we establish the following: (i) None of the principles AC$^\{\rm LO\}$ (AC for linearly ordered families of nonempty sets)—and hence AC$^\{\rm WO\}$ (AC for well-ordered families of nonempty sets)— $\{\rm DC\}(\{<\}\kappa )$ (where $\kappa $ is an uncountable regular cardinal), and “for every infinite set $X$, there is a bijection $f\colon X\rightarrow \lbrace 0,1\rbrace \times X$”, implies the statement “there exists a field $F$ such that every vector space over $F$ has a basis” in ZFA set theory. The above results settle the corresponding open problems from Howard and Rubin “Consequences of the axiom of choice:”, and also shed light on the question of Bleicher in “Some theorems on vector spaces and the axiom of choice” about the set-theoretic strength of the above algebraic statement. (ii) “For every field $F$, for every family $\mathcal \{V\}=\lbrace V_\{i\}\colon i\in I\rbrace $ of nontrivial vector spaces over $F$, there is a family $\mathcal \{F\}=\lbrace f_\{i\}\colon i\in I\rbrace $ such that $f_\{i\}\in F^\{V_\{i\}\}$ for all $ i\in I$, and $f_\{i\}$ is a nonzero linear functional” is equivalent to the full AC in ZFA set theory. (iii) “Every infinite-dimensional vector space over $\mathbb \{R\}$ has a norm” is not provable in ZF set theory.},
author = {Tachtsis, Eleftherios},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {choice principle; vector space; base for vector space; nonzero linear functional; norm on vector space; Fraenkel–Mostowski permutation models of $\{\rm ZFA\}+\lnot \{\rm AC\}$; Jech–Sochor first embedding theorem},
language = {eng},
number = {3},
pages = {285-309},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On certain non-constructive properties of infinite-dimensional vector spaces},
url = {http://eudml.org/doc/294791},
volume = {59},
year = {2018},
}

TY - JOUR
AU - Tachtsis, Eleftherios
TI - On certain non-constructive properties of infinite-dimensional vector spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2018
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 59
IS - 3
SP - 285
EP - 309
AB - In set theory without the axiom of choice (${\rm AC}$), we study certain non-constructive properties of infinite-dimensional vector spaces. Among several results, we establish the following: (i) None of the principles AC$^{\rm LO}$ (AC for linearly ordered families of nonempty sets)—and hence AC$^{\rm WO}$ (AC for well-ordered families of nonempty sets)— ${\rm DC}({<}\kappa )$ (where $\kappa $ is an uncountable regular cardinal), and “for every infinite set $X$, there is a bijection $f\colon X\rightarrow \lbrace 0,1\rbrace \times X$”, implies the statement “there exists a field $F$ such that every vector space over $F$ has a basis” in ZFA set theory. The above results settle the corresponding open problems from Howard and Rubin “Consequences of the axiom of choice:”, and also shed light on the question of Bleicher in “Some theorems on vector spaces and the axiom of choice” about the set-theoretic strength of the above algebraic statement. (ii) “For every field $F$, for every family $\mathcal {V}=\lbrace V_{i}\colon i\in I\rbrace $ of nontrivial vector spaces over $F$, there is a family $\mathcal {F}=\lbrace f_{i}\colon i\in I\rbrace $ such that $f_{i}\in F^{V_{i}}$ for all $ i\in I$, and $f_{i}$ is a nonzero linear functional” is equivalent to the full AC in ZFA set theory. (iii) “Every infinite-dimensional vector space over $\mathbb {R}$ has a norm” is not provable in ZF set theory.
LA - eng
KW - choice principle; vector space; base for vector space; nonzero linear functional; norm on vector space; Fraenkel–Mostowski permutation models of ${\rm ZFA}+\lnot {\rm AC}$; Jech–Sochor first embedding theorem
UR - http://eudml.org/doc/294791
ER -