A solution is to build up the meanings by approximation. The factorial function is a total function from ℕ to ℕ (defined everywhere in its domain), but we model it as a partial function. At the beginning, we start with the empty function (an empty set). Next, we add the ordered pair (0,1) to the function to result in another partial function that better approximates the factorial function. Afterwards, we add yet another ordered pair (1,1) to create an even better approximation.