Linear forms and axioms of choice

Marianne Morillon

Commentationes Mathematicae Universitatis Carolinae (2009)

  • Volume: 50, Issue: 3, page 421-431
  • ISSN: 0010-2628

Abstract

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We work in set-theory without choice ZF. Given a commutative field 𝕂 , we consider the statement 𝐃 ( 𝕂 ) : “On every non null 𝕂 -vector space there exists a non-null linear form.” We investigate various statements which are equivalent to 𝐃 ( 𝕂 ) in ZF. Denoting by ℤ 2 the two-element field, we deduce that 𝐃 ( ℤ 2 ) implies the axiom of choice for pairs. We also deduce that 𝐃 ( ℚ ) implies the axiom of choice for linearly ordered sets isomorphic with ℤ .

How to cite

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Morillon, Marianne. "Linear forms and axioms of choice." Commentationes Mathematicae Universitatis Carolinae 50.3 (2009): 421-431. <http://eudml.org/doc/33325>.

@article{Morillon2009,
abstract = {We work in set-theory without choice ZF. Given a commutative field $\mathbb \{K\}$, we consider the statement $\mathbf \{D\} (\mathbb \{K\})$: “On every non null $\mathbb \{K\}$-vector space there exists a non-null linear form.” We investigate various statements which are equivalent to $\mathbf \{D\} (\mathbb \{K\})$ in ZF. Denoting by $\mathbb \{Z\}_2$ the two-element field, we deduce that $\mathbf \{D\} (\mathbb \{Z\}_2)$ implies the axiom of choice for pairs. We also deduce that $\mathbf \{D\} (\mathbb \{Q\})$ implies the axiom of choice for linearly ordered sets isomorphic with $\mathbb \{Z\}$.},
author = {Morillon, Marianne},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Axiom of Choice; axiom of finite choice; bases in a vector space; linear forms; axiom of choice; axiom of finite choice; base in a vector space; linear form},
language = {eng},
number = {3},
pages = {421-431},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Linear forms and axioms of choice},
url = {http://eudml.org/doc/33325},
volume = {50},
year = {2009},
}

TY - JOUR
AU - Morillon, Marianne
TI - Linear forms and axioms of choice
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2009
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 50
IS - 3
SP - 421
EP - 431
AB - We work in set-theory without choice ZF. Given a commutative field $\mathbb {K}$, we consider the statement $\mathbf {D} (\mathbb {K})$: “On every non null $\mathbb {K}$-vector space there exists a non-null linear form.” We investigate various statements which are equivalent to $\mathbf {D} (\mathbb {K})$ in ZF. Denoting by $\mathbb {Z}_2$ the two-element field, we deduce that $\mathbf {D} (\mathbb {Z}_2)$ implies the axiom of choice for pairs. We also deduce that $\mathbf {D} (\mathbb {Q})$ implies the axiom of choice for linearly ordered sets isomorphic with $\mathbb {Z}$.
LA - eng
KW - Axiom of Choice; axiom of finite choice; bases in a vector space; linear forms; axiom of choice; axiom of finite choice; base in a vector space; linear form
UR - http://eudml.org/doc/33325
ER -

References

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