# Tagged Questions

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### Intuition for “the existence of a basis for every vector space is equivalent to the Axiom of Choice”?

Is there a intuitive way to understand "the existence of a basis for every vector space is equivalent to the Axiom of Choice"?
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### Easy visualizations of small countable ordinals

The ordinal number $\omega^2$ can be visualized as $\omega$-many copies of $\omega$. Likewise, the ordinal number $\omega^3$ can be visualized as $\omega^2$-many copies of $\omega$, arranged as ...
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### Cardinality of a vector space versus the cardinality of its basis

Let $V$ an infinite dimensional vector space. How to show that the cardinality of $V$ is the same of a basis of $V$? I saw this argument here link in the main answer (MathOverFlow).
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### A Vector Space is a Set - Axiom or Derivation?

I understand that structures with the properties of the real and complex numbers can be defined and derived from the axioms of ZFC set theory. But can a structure with the properties of a (possibly ...
What is the cardinality of a Hamel basis of $\ell_1(\mathbb{R})$? Is it deducible in ZFC that it is seemingly continuum? Does it follow from this that each Banach space of density \$\leqslant ...