Does there exist a undecidable infinite language with only a finite undecidable subset? I know that there's no such thing as a finitely sized undecidable language. However, does there exist an undecidable language where a finitely sized set of undecidable elements are 'hiding among' an infinite set of decidable elements?
In order to formalize this, let's build a deciding machine that answers $\text{YES}$, $\text{NO}$, or $\text{MAYBE}$. A correct decider for a language may always answer $\text{MAYBE}$, but if it answers $\text{YES}$ or $\text{NO}$ its answer must be correct. Such a decider can always be constructed for any (undecidable) language - in the worst case it simply always returns $\text{MAYBE}$.
Does there exist an undecidable language $L$ for which a correct decider exists that only answers $\text{MAYBE}$ for a finite number of elements of $L$?
Let $S$ be the set of elements on which the decider $T$ answers MAYBE. We can construct a new decider $T'$ which correctly decides $L$ as follows:

If the input is in $S$, answer the hardwired correct answer.
Otherwise, run $T'$.

Hence $L$ is decidable.
This actually shows that if $L$ is undecidable, then every decider for $L$ must make infinitely many mistakes.