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!