I was wondering how can one show that there exist $N$ real elements that is algebraically independent over $\mathbb{Q}$ for any $N$? (I was thinking perhaps Lindemann–Weierstrass theorem can be used. If we can find $N$ real linearly independent over $\mathbb{Q}$ algebraic numbers then the statement follows by Lindemann–Weierstrass. but I wasn't sure how to find such $N$ real algebraic numbers...)
Any comments and suggestions appreciated!
If you alread have $\alpha_1,\ldots,\alpha_{N-1}$ that are algebriacally independent, then $F=\Bbb Q(\alpha_1,\ldots,\alpha_n)$ is still countable, hence $F[X]$ is countable, hence there are only countably many real numbers that are algebraic ocer $F$. Pick one of the uncountably many others as your $\alpha_N$.