"Suppose the set T of number which cannot be written down is nonempty. Choose some t in T. As t is a written form an element of T, it has been written down, and so cannot be in T. Contradiction."
In math, you have to specify how you are writing things down - what language. If the language you are using is consistent, then there are, by necessity, true things which cannot be written in that language. For every two consistent languages, there is another language that can say by itself everything that either of the two says on their own. No matter how sophisticated you make your system, there will always be things you cannot say, or else you will be struck by inconsistencies.
Whether an inconsistent language like English can say anything that can be said (admitting ridiculous things like infinite length descriptions) or not, I have no idea.
If we allow infinite length statements, then for every true thing, there is a language of some sort which can note it - simply construct a language with the truth of the thing as one of its axioms. Then it can most certainly state it. In this sense, there is no set of true things (such as the existence of a number satisfying a property) that cannot be said by any