Let $k$ be a field, let $V$ be a vector space over $k$, and let $B,C \subset V$. For any subset $T \subset V$, define $\operatorname{span}_k(T) \subset V$ to be the set of all finite $k$-linear combinations of elements of $T$. Assume the usual set-theoretic cardinal comparison facts for sets: the Cantor-Bernstein theorem, $|\mathcal{P}_{\mathrm{fin}}(X)| = |X|$ for every infinite set $X$, and $\kappa \cdot \aleph_0 = \kappa$ for every infinite cardinal $\kappa$. If $B$ and $C$ are bases of $V$ over $k$, then there exists a bijection $B \to C$. Equivalently, $|B| = |C|$.