It's well-known that there exists some real numbers that cannot be defined in a string of finite length (Berry's paradox). However, why can the set of all real numbers be defined?
My gut feelings are
$1$. the definition has some nonconstructive descriptions, like the supremum and infimum principle;
$2$. or we are just talking about computable numbers?
I'm not familiar with real analysis & mathematical logic so there might be something I overlooked.