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How can I prove that the basis of the vector space $\mathbb{R}$ over $\mathbb{Q}$ is uncountable.

By vector space $\mathbb R$ over $\mathbb Q$ we mean $\mathbb R$ with addition and scalar multiplication as described, for example, in this post: Prove $\mathbb R$ vector space over $\mathbb Q$

A set $B$ is a basis of $\mathbb R$ over $\mathbb Q$ if every real number $x$ can be expressed uniquely as $$x = q_1b_1+\dots+q_nb_n,$$ where $q_1,\dots,q_n\in\mathbb Q$ and $b_1,\dots,b_n\in B$.

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HINT: Suppose that $B$ is a countable base for $\Bbb R$ over $\Bbb Q$. Then every real number can be written in the form

$$q_1b_1+q_2b_2+\ldots+q_nb_n$$

for some positive integer $n$ and $n$-tuples $\{b_1,b_2,\dots,b_n\}\in B^n$ and $\langle q_1,q_2,\dots,q_n\rangle\in\Bbb Q^n$.

  1. How many positive integers are there?
  2. How many elements does $\Bbb Q^n$ have?
  3. How many elements does $B^n$ have?
  4. How many combinations of $n\in\Bbb Z^+$, $\{b_1,b_2,\dots,b_n\}\in B^n$ and $\langle q_1,q_2,\dots,q_n\rangle\in\Bbb Q^n$ are there altogether?
Brian M. Scott
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    Aren't you assuming that B is finite? B can be countably infinite too. – LoneStar Sep 19 '19 at 02:13
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    @LoneStar: (I must have missed this in 2019; I’ll answer now for the benefit of future readers.) No, I’m assuming only that $B$ is countable. That is enough to ensure that all four questions have the same answer: countably many. – Brian M. Scott Aug 23 '24 at 17:16