The top rated answer to Why, really, is the Halting Problem so important? lists a few examples for a noncomputable problem. However, these mostly involve an infinite search space. Are there noncomputable problems with a finite search space? If not, why not?
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Depends on what exactly you have in mind.
If the problem is to find an element with decidable (!) properties among a recursively enumerable set $A$ (of finitely represented elements), then yes: if $A$ is finite, the problem is trivially computable. Just check every element.
If $A$ is finite but unknown, it's more interesting but still: the problem is computable, since there is an algorithm that checks the every element in $A$, even though we can't point it out. See this for an example.
If the identifying property is undecidable, then the problem may be¹ uncomputable. For instance, the search space for the question "Does TM M halt on input x?" is finite -- just two possible answers -- but the problem is (in)famously undecidable.
- It's possible that the property as written down is not needed to identify the correct element.
Raphael
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