| Title:
|
Linear forms and axioms of choice (English) |
| Author:
|
Morillon, Marianne |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
50 |
| Issue:
|
3 |
| Year:
|
2009 |
| Pages:
|
421-431 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $\mathbb Z$. (English) |
| Keyword:
|
Axiom of Choice |
| Keyword:
|
axiom of finite choice |
| Keyword:
|
bases in a vector space |
| Keyword:
|
linear forms |
| MSC:
|
03E25 |
| MSC:
|
15A03 |
| idZBL:
|
Zbl 1212.03034 |
| idMR:
|
MR2573415 |
| . |
| Date available:
|
2009-09-23T21:35:01Z |
| Last updated:
|
2013-09-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134914 |
| . |
| Reference:
|
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| Reference:
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