How to show basis of the vector space $\mathbb{R}$ over $\mathbb{Q}$ is uncountable.

Question

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 $b_1,\dots,b_n\in B$.

If the basis were counatable, so would be the space. We yould have $\mathbb R\cong \mathbb Q[X]$. — Hagen von Eitzen, Oct 27 '12 at 16:30
Related: http://math.stackexchange.com/questions/583601/take-mathbbr-as-a-vector-space-over-mathbbq-then-a-basis-for-mathbb — Martin Sleziak, Feb 16 '15 at 09:46
Since the question seems to have been abandoned by the OP and it received some close votes, I tried to improve the question. (I think that it is useful to have post about this topic on the site.) — Martin Sleziak, Feb 16 '15 at 09:52

score 4 · Accepted Answer · answered Oct 27 '12 at 16:36

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 $\langle q_1,q_2,\dots,q_n\rangle\in\Bbb Q^n$.

How many positive integers are there?
How many elements does $\Bbb Q^n$ have?
How many elements does $B^n$ have?
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?

Aren't you assuming that B is finite? B can be countably infinite too. — LoneStar, Sep 19 '19 at 02:13

How to show basis of the vector space $\mathbb{R}$ over $\mathbb{Q}$ is uncountable.

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