Is there a "natural" undecidable language? Is there any "natural" language which is undecidable?

by "natural" I mean a language defined directly by properties of strings, and not via machines and their equivalent. In other words, if the language looks like
$$ L = \{ \langle M \rangle \mid \ldots \}$$
where $M$ is a TM, DFA (or regular-exp), PDA (or grammar), etc.., then $L$ is not natural. However $L = \{xy \ldots \mid x \text{ is a prefix of y} \ldots \}$ is natural.
Since you wanted "strings", I mention the classic one: Post Correspondence Problem.