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Results tagged with vector-spaces
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user 7206
10
votes
Accepted
Properties of vector spaces without AC
(I completely revamped my answer, the previous version (link) had a consistency result with a particular example, this feels much better as it establishes a full equivalence result instead.)
The answ …
- 39.9k
6
votes
Accepted
Linear space with (Hamel) basis and the axiom of choice
No. It is not possible.
Suppose that $V$ is a specified vector space, then it is consistent that the axiom of choice fails very far above $V$ in the hierarchy of sets (the von Neumann hierarchy). In …
- 39.9k
10
votes
Accepted
Relation between well-orderings of $\mathbb{R}$, and bases over $\mathbb{Q}$
The problem is generally open. However, recently Liuzhen Wu, Liang Yu, Ralf Schindler and Mariam Beriashvili posted a preprint in which they prove the consistency of the existence of a Hamel basis and …
- 39.9k
13
votes
Accepted
Axiom(s) of choice and bases of vector spaces
There is no fully elementary proof that you are looking for. The reason is that the axiom of regularity is needed in these proofs. Multiple Choice does not imply Choice without it, and the only proofs …
- 39.9k
15
votes
Accepted
Examples of vector spaces with bases of different cardinalities
This is not a very thoroughly studied problem. So to start from the end, there is no standard procedure for this sort of construction. We know of one, it can maybe be adapted slightly to get a mildly …
- 39.9k
5
votes
Sets of cardinalities of bases without choice
Sizes of bases of vector spaces without the axiom of choice shows that assuming $\sf BPI$ we have that every two bases have the same cardinality. This means that $BS(V)$ is either empty, or a singleto …
- 39.9k