Undecidable unary languages (also known as Tally languages)

An exercise that was in a past session is the following:


Prove that there exists an undecidable subset of $\{1\}^*$

This exercise looks very strange to me, because I think that all subsets are decidable.

Is there a topic that I should read to find a possible answer?

Note that $\{1\}^$ is isomorphic to $\Bbb N$. There are uncountably many subsets of both $\{1\}^$ and $\Bbb N$.

Perhaps you are confused with the fact that there are only countably many finite subsets of $\{1\}^*$ (and $\Bbb N$).

StackExchange Computer Science Q#19329, answer score: 6