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.


There are many examples but here are a few:

  • The set of true sentences in the language of arithmetic is undecidable.

  • The set of provable sentences in set theory (ZFC) is undecidable.

  • The set of Diophantine equations which have solutions is undecidable.


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