TITLE: A question about a method that shows $\mathbb{R} $ not finite dimensional.
QUESTION [5 upvotes]: Upon looking at methods that show $\mathbb{R}$ is not finite dimensional over $\mathbb{Q}$ I came across a method mentioned here by the user Bill Dubuque, he took a set of vectors of the form $\log(p)$ where $p$ is prime and showed that the set is independent, but in his proof he only takes $n$-primes. So my questions are:
The set of these logarithms is infinite , why did he only use $n$ primes? 
Why does this show that $\mathbb{R}$ is not finite dimensional?
For the second question I'm not sure but I think it is because no matter what $n$ is the set is independent, but I'm not sure this can be extended to $n= \infty$.

REPLY [4 votes]: To show that $\{\log p_i\mid i\in \Bbb N\}$ is linearly independent over $\Bbb Q$, you would just need to show that every nontrivial linear combination of its elements is nonzero.
So, select finitely many elements of the set to combine. This set must have a highest index $\log p_i$, so without loss of generality, we can add all the $\log p_j$ into this linear combination that are absent simply by giving them the coefficient zero. Thus whatever finite linear combination you start with, you can always change it to include the first $n$ prime-logs.
The conclusion is that this combination (equal to the original combination) is nonzero, and so $\{\log p_i\mid i\in \Bbb N\}$ is LI over $\Bbb Q$.