A decision problem is a yes-or-no question on an infinite set of inputs. It is traditional to define the decision problem as the set of possible inputs together with the set of inputs for which the answer is yes.
These inputs can be natural numbers, but can also be values of some other kind, like binary strings or strings over some other alphabet. The subset of strings for which the problem returns "yes" is a formal language, and often decision problems are defined as formal languages.