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choice principle; vector space; base for vector space; nonzero linear functional; norm on vector space; Fraenkel--Mostowski permutation models of ${\rm ZFA}+\neg{\rm AC}$; Jech--Sochor first embedding theorem
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 $f\colon X\rightarrow\{0,1\}\times X$", 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 $\mathcal{V}=\{V_{i}\colon i\in I\}$ of nontrivial vector spaces over $F$, there is a family $\mathcal{F}=\{f_{i}\colon i\in I\}$ 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.
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