No surjective function exists from names to real numbers (diagonalization). With any naming scheme, some unnamed real numbers always remain.

On the other hand, given any real number, I can name it. I'll run out of unique names, since no injective function exists from real numbers to names.

So, some unnamed real numbers will always remain (non-constructively), I can make real numbers that escape a naming scheme (constructively), and no unnameable real numbers exist.