  We prove that the strong Bombieri-Lang conjecture for $\mathbb{Q}$ implies it
for fields finitely generated over $\mathbb{Q}$. We prove that the geometric
Lang conjecture plus the weak Bombieri-Lang conjecture over $\mathbb{Q}$ imply
the strong Bombieri-Lang conjecture for fields finitely generated over
$\mathbb{Q}$. Thus we reduce the arithmetic part of the conjectures to
$\mathbb{Q}$ removing the need, for arithmetic purposes, of Lang's conjecture
for function fields.
