Real numbers as a $\mathbb{Q}$-linear space [duplicate]

Let's consider the real numbers as a $\mathbb{Q}$-linear space and denote it by $\mathbb{R}_{\mathbb{Q}}$. It was proved here that $\dim \mathbb{R}_{\mathbb{Q}} > \aleph_0$. I suspect that the dimension is actually $\mathfrak c$. But how could we prove it's really $\mathfrak c$ ?

marked as duplicate by Dietrich Burde, yoknapatawpha, Dan, Silvia Ghinassi, ShaileshFeb 18 '16 at 0:07

• If $\kappa$ is an infinite cardinal then a set of cardinality $\kappa$ has only $\kappa$ finite subsets... – David C. Ullrich Feb 17 '16 at 13:49