| Title:
|
On certain non-constructive properties of infinite-dimensional vector spaces (English) |
| Author:
|
Tachtsis, Eleftherios |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
59 |
| Issue:
|
3 |
| Year:
|
2018 |
| Pages:
|
285-309 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
choice principle |
| Keyword:
|
vector space |
| Keyword:
|
base for vector space |
| Keyword:
|
nonzero linear functional |
| Keyword:
|
norm on vector space |
| Keyword:
|
Fraenkel--Mostowski permutation models of ${\rm ZFA}+\neg{\rm AC}$ |
| Keyword:
|
Jech--Sochor first embedding theorem |
| MSC:
|
03E25 |
| MSC:
|
03E35 |
| MSC:
|
15A03 |
| MSC:
|
15A04 |
| idZBL:
|
Zbl 06940871 |
| idMR:
|
MR3861553 |
| DOI:
|
10.14712/1213-7243.2015.258 |
| . |
| Date available:
|
2018-09-10T12:08:15Z |
| Last updated:
|
2020-10-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147398 |
| . |
| Reference:
|
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| Reference:
|
[2] Blass A.: Existence of bases implies the axiom of choice.Axiomatic Set Theory, Contemp. Math., 31, Amer. Math. Soc., Providence, 1984, pages 31–33. Zbl 0557.03030, MR 0763890, 10.1090/conm/031/763890 |
| Reference:
|
[3] Bleicher M. N.: Some theorems on vector spaces and the axiom of choice.Fund. Math. 54 (1964), 95–107. Zbl 0118.25503, MR 0164899, 10.4064/fm-54-1-95-107 |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
[7] Howard P., Tachtsis E.: No decreasing sequence of cardinals.Arch. Math. Logic 55 (2016), no. 3–4, 415–429. MR 3490912, 10.1007/s00153-015-0472-5 |
| Reference:
|
[8] Howard P., Tachtsis E.: On infinite-dimensional Banach spaces and weak forms of the axiom of choice.MLQ Math. Log. Q. 63 (2017), no. 6, 509–535. MR 3755261, 10.1002/malq.201600027 |
| Reference:
|
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| Reference:
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| Reference:
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| Reference:
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| Reference:
|
[13] Rubin H., Rubin J. E.: Equivalents of the Axiom of Choice, II.Studies in Logic and the Foundations of Mathematics, 116, North-Holland Publishing, Amsterdam, 1985. MR 0798475 |
| . |
