Undecidable languages
Here is a more general statement:


  A language $L$ has an undecidable subset iff $L$ is infinite.


For the proof, note first that if $L$ is finite then all its subsets are finite and so decidable. Conversely, if $L$ is infinite then it has uncountably many subsets. Since there are only countably many decidable languages, some subset of $L$ (indeed, most subsets of $L$) must be undecidable.