In mathematical practice, we find at least two notions of undecidability. On the one hand, a decision problem can be computably undecidable, meaning that there is no computable procedure that correctly solves all instances of the problem. For example, the halting problem is undecidable, as is the tiling problem and the diophantine equation problem. On the other hand, mathematicians say that a statement is undecidable in a theory T, when the theory neither proves nor refutes the statement.
